diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzezzu" "b/data_all_eng_slimpj/shuffled/split2/finalzzezzu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzezzu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe abundance of Li has attracted much attention, especially since the Li gap\nhas been discovered in the Hyades for stars with \\teff $\\sim 6600$~K. In the\ncontext of radiative diffusion, it is interesting to examine the atmospheric\nabundance of lithium in stars where such a mechanism is known to be at work from\nthe abundances of other elements, such as calcium, i.e. in the Am stars.\nSuch studies have been carried out especially by Burkhart \\& Coupry \\citep{bc91}\nand Burkhart et al. \\citep{bcfg05}.\nTheir conclusion was that in general, the Li abundance\nof Am stars is close to the cosmic value ($\\log N(Li)\\sim 3.0$ in the scale\nwhere $\\log N(H)= 12.0$), although a small proportion of them are deficient.\nThe latter seem in general to be either evolved stars or, as recently suggested\nby Burkhart et al. \\citep{bcfg05}, to lie on the red side of the Am domain,\namong the $\\rho$ Puppis--like stars.\n\nIn this poster, we present Li abundances obtained for 31 Am stars and 36 normal\nA and F stars in the field, all having Hipparcos parallaxes. This sample had\nbeen defined before the Hipparcos era, on purely photometric criteria, but with\nthe purpose of testing how far the Li abundance depends on the evolutionary\nstate, i.e. on the surface gravity $\\log g$. The Hipparcos data which became\navailable later showed that the photometric luminosity calibrations of Am stars\nwere not very satisfactory (North et al. 1997), but allowed to determine\n$\\log g$ in a more fundamental way. Furthermore, the sample has the advantage\nof presenting no bias against large rotational velocities.\n\n\n\\section{Observations and analysis}\nAll stars were observed at OHP with the Aur\\'elie spectrograph attached to the\n1.5m telescope, in April 1993 and in October 1993 and 1994. The grating No 7 was\nused, giving a resolving power $R=40000$ in the spectral range $6640-6760$~\\AA .\nThe typical exposure times were between 40 and 60 minutes, the resulting\nsignal-to-noise ratio being between 250 and 400. The spectra were\nreduced during the observing runs with the IHAP package, and were later\nnormalized to the continuum in an interactive way.\n\nThe analysis was made by comparison of the observed spectra with synthetic ones\nconvoluted with an assumed gaussian instrumental profile and with an appropriate\nrotational profile. The Synspec code (Hubeny et al. 1995) and Kurucz model\natmospheres were used to produce the synthetic spectra. The line parameters were\ntaken from Kurucz's $gfiron$ list, except of course the parameters for the Li\ndoublet. The effective temperatures were computed from Geneva photometry, while\nthe surface gravities were computed from the Hipparcos parallaxes, by combining\nthem with theoretical evolutionary tracks from Schaller et al. \\citep{ssmm92},\nas explained by North \\citep{n98}, assuming standard evolution.\nThe microturbulent velocity was either computed from the formula proposed by\nEdvardsson et al. \\citep[eqn 9]{e93}, for \\teff $< 7000$~K, or estimated from the\nFig.~1 of Coupry \\& Burkhart \\citep{cb92}, for \\teff $\\geq 7000$~K.\nThe abundance of\nFe, Ca and a few other elements (in cases of sharp lined stars) were first\nestimated by visual fits. Then, the Li abundance and the projected rotational\nvelocities were obtained by minimizing the $\\chi^2$ between observed and\nsynthetic spectra having various values of these parameters.\n\\begin{figure}\n \\includegraphics[width=10cm]{north_hr.eps}\n \\caption{HR diagram of the Am (black dots) and normal (white dots) stars of\nour sample. Black triangles (with error bars typical of the whole\nsample) are stars from Burkhart et al. (2005, Table~3) not in\ncommon with our sample. The error bars were drawn assuming a $\\pm 200$~K error\non \\teff and include, on the vertical axis, the parallax error of Hipparcos.\n}\n\\label{hr}\n\\end{figure}\n\\section{Results}\nFig.~\\ref{hr} shows the distribution of Am stars (full dots) and of normal A-F stars\n(open dots) in the HR diagram. Evolutionary tracks and isochrones from\nSchaller et al. \\citep{ssmm92} are shown for 4 masses ($1.5$ to $2.5~M_\\odot$)\nand for 3\nages ($\\log t = 8.7$ to $9.3$) respectively. The stars are well distributed\non the whole main sequence band. The lack of Am stars below \\teff $\\sim 7000$~K\nis the well-known limit due to the onset of convection.\n\nFig.~\\ref{LiTelg} (left) shows the lithium abundance as a function of \\teff\nfor Am stars (full dots) and for normal A--F stars (open dots).\nThe most striking feature of this diagram is\nthe bimodal distribution of the Li abundance for \\teff $\\lesssim 7500$~K, which\nis reminiscent of a similar distribution of F-type dwarfs in the range $5900\n<$ \\teff $< 6600$~K reported by Lambert et al. \\citep[Fig.~4]{lhe91}.\nThus, our data\ncomplement that of Lambert et al. as well as the larger sample of Chen et al.\n\\citep{cnbz01} by extending the results to higher \\teff. We have\nverified that duplicity cannot account for the low apparent Li abundances (even\nthough this might hold for some isolated cases). Restricting the diagram to\nthose stars with\n\\vsini $< 80$~\\kms, the upper branch almost disappears (there are only two\nnormal stars left around \\teff $\\sim 6500$~K), while the lower one remains\nintact. This is related to the fact that the upper branch is populated only with\nnormal stars, which rotate more rapidly than the Am stars, while the lower\nbranch is a mix of normal and Am stars. Thus, below $7500$~K, all Am stars of\nour sample are Li deficient. The black triangles refer to the 4 stars of\nBurkhart et al. \\citep[their Table~3]{bcfg05} which are not common to our\nsample. Their positions are in perfect agreement with the general picture.\n\nFig.~\\ref{LiTelg} (right) displays the Li abundance as a function of surface gravity. There is\nno strong trend, but one can notice that those stars (either Am or normal) which\nare strongly deficient in Li are {\\bf all} at least slightly evolved. There is one\nunevolved star (HD 18769) for which only an upper limit to its Li abundance\ncould be obtained, but this is due to its high \\teff ($8420$~K) and moderately\nbroad lines, and the upper limit is close to the ``cosmic'' Li abundance, so\nthis is not a significant exception. Thus, we confirm the suggestion made by\nBurkhart \\& Coupry that Li-deficient Am stars are evolved objects, although it\nseems that all evolved Am stars are not necessarily deficient.\n\\begin{figure}\n \\includegraphics[width=10.3cm]{north_LiTelg.eps}\n \\caption{{\\bf Left:} Li abundance (on the scale $\\log N(H)=12$) of Am\nstars (black\ndots) and normal A--F-type stars (white dots) versus effective temperature.\nUpper limits to the Li abundance are indicated by vertical arrows.\nBlack triangles are from Burkhart et al. \\citep{bcfg05}.\n{\\bf Right:} Li abundance versus surface gravity derived from Hipparcos\nparallaxes. The leftmost arrow refers to the Am star HD 18769, which has\n\\teff $=8420$~K and \\vsini $= 46$~\\kms, so that only an upper limit\nto its Li abundance can be obtained. The typical error on $\\log g$ is $0.1$~dex,\nwhile that on $\\log N(Li)$ vary from better than $0.1$~dex to more than\n$0.3$~dex, depending on \\teff and \\vsini.}\n\\label{LiTelg}\n\\end{figure}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\n\n\nThere is a voluminous literature on second order analysis of distribution functions $F_N(z) = P(Z_N\\leq z)$ of statistics $Z_N = \\zeta_N(X_1,X_2,\\dots, X_N)$ that are functions of i.i.d. random variables $X_1,X_2,...$. The results obtained are generally refinements of the central limit theorem. Suppose that $Z_N$ is asymptotically standard normal, that is $\\sup_z |F_N(z)-\\Phi (z)|\\rightarrow0 $ as $N\\rightarrow \\infty$, where $\\Phi $ denotes the standard normal distribution function. Then second order results are concerned with the speed of this convergence, or with attempts to increase this speed by replacing the limit $\\Phi $ by a series expansion $\\Psi_N$ that provides a better approximation. Results of the first kind are called theorems of Berry-Esseen type and assert that for all $N$,\n$$\\sup_z|F_N(z)-\\Phi(z)|\\leq CN^{-\\frac12},$$\nwhere $C$ is a constant that depends on the particular statistic $Z_N$ and the\ndistribution of the variables $X_i$, but not on $N$. Such results are often\nvalid under mild restrictions such as the existence of a third moment of\n$X_i$. The original Berry-Esseen theorem dealt with the case where $Z_N$ is a\nsum of i.i.d. random variables, \n\\citet*{Esseen:1942},\n\\citet*{Berry:1941}. \nFor a more general version see \n\\citet*{vanZwet:1984}. \n\n\nResults of the second kind concern so-called Edgeworth expansions. These are series expansions such as \n\\begin{equation} \n \\begin{split}\n &\\Psi_{ N,1}(z) = \\Phi(z)+\\varphi(z)N^{-\\frac12} Q_1(z), \\quad \\text{or} \\\\ \n &\\Psi_{N,2}(z) =\\Phi(z)+\\varphi(z)\\left[ N^{-\\frac12} Q_1(z)+ N^{-1}Q_2(z)\\right],\n \\end{split}\n\\end{equation} \n\n\\noindent\nwhere $ \\varphi $ is the standard normal density and $Q_1$ and $Q_2$ are polynomials depending on low moments of $X_i$ . One then shows that \n\\begin{equation} \n \\begin{split}\n &\\sup_z|F_N(z)-\\Psi_{N,1}(z)|\\leq CN^{-1}, \\quad \\text{or} \\\\ \n &\\sup_z|F_N(z)-\\Psi_{N,2}(z)|\\leq CN^{-\\frac32} .\n \\end{split}\n\\end{equation} \n\n\\noindent\nFor this type of result the restrictions are more severe. Apart from moment\nassumptions one typically assumes that $Z_N$ is not a lattice random\nvariable. For the case where $Z_N$ is a sum of i.i.d. random variables a good\nreference is \n\\citep[chap. XVI]{Feller:1965\/2}. \nThere are numerous papers devoted to special types of statistics. For a\nsomewhat more general result we refer to \n\\citet*{Bickel-Goetze-vanZwet:1986}\n and \n\\citet*{bentkus-goetze-vanzwet:1997} .\n\nFor the case where $Z_N$ assumes its values on a lattice, say the integers, an alternative approach it to generalize the local central limit theorem and provide an expansion for the point probabilities $P(Z_N=z)$ for values of $z$ belonging to the lattice. A typical case is the binomial distribution for which local expansions are well known. It is obvious that for the binomial distribution one can not obtain Edgeworth expansions as given in $(1.1)$ for which $(1.2)$ holds. The reason is that out of the $N$ possible values for a binomial $(N,p)$ random variable, only $cN^{\\frac12}$ values around the mean $Np$ really count and each of these has probability of order $N^{-\\frac12}$. Hence the distribution function has jumps of order $N^{-\\frac12}$ and can therefore not be approximated by a continuous function such as given in $(1.1)$ with an error of smaller order than $N^{-\\frac12}$. \n\nIn a sense the binomial example is an extreme case where the ease of\nthe approach through local expansions for $P(Z_N=z)$ precludes the one through\nexpansions of Edgeworth type for $P(Z_N\\leq z)$. In \n\\citet*{Albers-Bickel-vanZwet:1976}\n the authors found somewhat to their surprise that for the Wilcoxon statistic which is a pure lattice statistic, an Edgeworth expansion with remainder $O(N^{-\\frac32})$ for the distribution function is perfectly possible. In this case the statistic ranges over $N^2$ possible integer values, of which the central $N^{\\frac32}$ values have probabilities of order $N^{-\\frac32}$ so that one can approximate a distribution function with such jumps by a continuous one with error $O(N^{-\\frac32})$.\n \nOn the basis of these examples one might guess that the existence of an Edgeworth expansion with error $O(N^{-p})$ for the distribution function $F_N(z) = P(Z_N\\leq z)$ would merely depend on the existence of some moments of $Z_N$ combined with the requirement that $F_N$ does not exhibit jumps of large order than $N^{-p}$. But one can envisage a more subtle problem if $F_N$ would assign a probability of larger order than $N^{-p}$ to an interval of length $N^{-p}$. Since Edgeworth expansions have bounded derivative, this would also preclude the existence of such an expansion with error $O(N^{-p})$.\n \nLittle seems to be known about the case where $Z_N$ has a discrete but non-lattice distribution. Examples abound if one considers a lattice random variable with expectation $0$ and standardized by dividing by its sample standard deviation. As a simple example, one could for instance consider Student's $t$-statistic $\\tau_N = N^{-1\/2}\\sum_i X_i \/ \\sqrt{\\sum_i\\left( X_i -m \\right)^2\/(N-1) }$ with $m= \\sum_i X_i\/N $ and $X_1,X_2,\\dots$ i..i.d. random variables with a lattice distribution. Since we are not interested in any particular statistic, but merely in exploring what goes on in a case like this, we shall simplify even further by deleting the sample mean m and considering the statistic\\smallskip\n\n\\noindent\n\\begin{equation} \nW_N = \\sum_{i=1}^N \\frac{X_i}{\\sqrt{\\sum_{i=1}^N X_i^2}} ,\n\\end{equation}\n\n\\noindent\nwith \\smallskip\n\n\\noindent\n\\begin{equation}\n X_1,X_2,\\dots i.i.d.\\mbox{ with } P(X_i=-1)= P(X_i=0)= P(X_i=1)= \\frac13 .\n\\end{equation}\n\n\nWe should perhaps point out that for $w>0$\n\\begin{align*}\nP(0<\\tau_N\\leq w)&=P\\left(00$, $\\Lambda_N(w)$ is bounded and $N^{-1\/2}\\Lambda_N(w) = O(N^{-1\/2})$ uniformly in $w$. At first sight there is a striking similarity between the expansion $\\Psi_N$ in Theorem 1.1 and the two term Edgeworth expansion $\\Psi_{N,1}$ in (1.1). However, the term $\\phi(z)N^{-1\/2}Q_1(z)$ of order $O(N^{-1\/2})$ in the Edgeworth expansion is a skewness correction that vanishes for a symmetric distribution $F_N$. As we are dealing with a symmetric case, such a term is not present and for the continuous case the Edgeworth expansion with remainder $O(N^{-1})$ is simply $\\Phi(z)$. The origin of the term $N^{-1\/2}\\Lambda_N(w)$ is quite different. It arises from the fact that we are approximating a discrete distribution function by a continuous one, and as such it is akin to the classical continuity correction.\n\nTo make sure that the term $N^{-1\/2}\\Lambda_N(w)$ is not of smaller order than $N^{-1\/2}$, we shall bound $|\\Lambda_N(w)|$ from below by the absolute value \nof the following series. Assume that $N$ is divisible by $3$ and let\n \\begin{equation}\n \\begin{split}\n\\lambda_N(w):=& \\sqrt{\\frac 3 2}\\4 \\varphi(w)\\sum_{k=1}^{M} \\frac 1 {\\pi \\4 k} \nf_{N,k}\\exp\\bigl(- \\frac{\\pi^2}6\\4 k^2\\4 w^2\\4)\\4 \n\\sin\\bigl(2 \\4 \\pi\\4 \\4 k \\4 w\\4 \\sqrt{\\frac {2 N} 3}\\bigr)\\\\\n& \\, \\quad + O\\bigl(N^{-1\/2}(\\log N)^5\\bigr), \\quad M := \\lfloor \\log N \\rfloor ,\n \\end{split}\n \\end{equation}\nwhere $f_{N,k}=1 + O((k\/M)^2)$ is defined in $(3.9)$. Thus $\\lambda_N(w)$ \nis a rapidly converging Fourier series, (illustrated in\nFigure 1. below) the modulus of which is larger than a positive constant $c(w)>0$,\nprovided that $4\\4 w\\4 \\sqrt{\\frac{2 N} 3} $ is an odd integer.\n\n\n\n\\begin{figure}[H] \n\\psfig{file=oscillatory.eps,width=18cm, height=4cm}\n\\caption{$\\lambda_{100}(w)$:\\,\\, $0.05 \\le w \\le 2.34,\\,\\, M=10$, $f_{100,k}:=\\exp[-(k\/M)^{2\/3}]$\\label{fig1}} \n\\end{figure} \n Hence, we shall prove\n\\noindent\n{\\bf Theorem 1.2.} {\\it For any $N$ divisible by $3$, we have} \n\\begin{equation}\n \\sup_{w>0} |F_N(w)-\\Phi(w)| \\ge \\,\\,\n\\sup_{w\\ge 1}N^{-\\frac 12}|\\lambda_N(w)| + O\\bigl(N^{-1} (\\log N)^5\\bigr)\n>\\,\\, \\frac c {\\sqrt N},\n\\end{equation}\n{\\it for some absolute constant $c>0$.} \n\n\\bigskip\\noindent\nThe proof of Theorem 1.1 is given in Section 2. In Section 3 we investigate \nthe oscillatory part of $\\Psi_N$ in (1.6), relating it to the\nFourier series $\\lambda_N(w)$ above and thus proving Theorem 1.2.\n\n\\noindent\n{\\bf Acknowledgment.}\n The authors would like to thank G.Chistyakov for a careful reading of the\n manuscript\n and Lutz Mattner for his comments on the current ArXiv version.\n \n\n\\section{Proof of Theorem 1.1}\n\nThe event $W_N = 0$ occurs iff $D_N = 0$. Let $Z_1,Z_2,\\dots, Z_N$ be i.i.d. random variables assuming the values $0$, $-1$ and $+1$, each with probability $\\frac 13$. Then $D_N$ is distributed as $ \\sum Z_i$, which has mean $0$ and variance $\\frac{2N}{3}$. By the local central limit theorem $P(\\sum Z_i =0) \\sim (2\\pi)^{-\\frac12} \\left(\\frac{2N}{3} \\right)^{-\\frac12} = \\sqrt{\\frac{3}{4\\pi N}}$ which proves the first statement of Theorem 1.1. Because the distribution of $W_N$ is symmetric about the origin, this implies that in the remainder of the proof we only need to consider positive values of $W_N$. Hence we suppose that $w>0$ throughout and this implies that we need only be concerned with positive values of $D_N$ also.\n\nHoeffding's inequality ensures that for all $N\\geq 2$,\n \\[P\\left(|D_N|\\geq \\sqrt{6N \\log N} \\right)\\leq \\frac{2}{N^{3}} \\]\nand \n\\[P\\left(|T_N -2N\/3|\\geq \\sqrt{2 N \\log N}\\right) \\leq \\frac{2}{N^{3}}.\\] \n\\noindent\nSince the joint distribution of $T_N$ and $D_N$ assigns positive probability to at most $N^2$ points and events with probability $O(N^{-1})$ are irrelevant for the remainder of the proof, we may at any point restrict attention to values $D_N=d$ and $T_N=t$ with $|d| \\leq t$ and satisfying \\smallskip\n\n\\noindent\n \\begin{equation}\n |d|< \\sqrt{6N\\log N} \\quad \\mbox{ and }\\quad \\left|t-\\frac{2N}{3}\\right|< \\sqrt{2N\\log N}. \n \\end{equation}\n\nFor positive integer $m\\leq n$ we have\n\\begin{eqnarray*}\n P(D_N =2m, T_N =2n)&=& P(S_N =m+n, T_N =2n) \\\\\n &=& \\frac{N!}{3^{N}(n+m)! (n-m)! (N-2n)!} .\n\\end{eqnarray*}\n\\noindent\nIf $d=2m$ and $t=2n$ satisfy $(2.1)$, then $(n+m)$, $(n-m)$ and $(N-2n)$ are of exact order $N$ and we may apply Stirling's formula to see that\n$$ P(D_N=2m,T_N =2n)$$ \n$$ = \\frac{N^{N+\\frac12} \\left(1+O\\left(\\frac 1N\\right)\\right) }{ 2\\pi 3^{N} (n+m)^{(n+m+\\frac12)}(n-m)^{(n-m+\\frac12)}(N-2n)^{(N-2n+\\frac12)}} $$\n$$ = \\frac{3^{\\frac 32} \\left(1+O\\left(\\frac 1N\\right)\\right) } { 2\\pi N\\left(\\frac{ 3(n+m)}{N}\\right)^{(n+m+\\frac12)}\\left( \\frac{3(n-m)}{N}\\right)^{(n-m+\\frac12)} \\left(\\frac{3(N-2n)}{N}\\right)^{(N-2n+\\frac12)}} $$\n$$ = \\frac{3^{\\frac32} \\left(1+O\\left( \\frac 1N \\right)\\right)} { 2\\pi N } \\, \\exp \\Bigg\\{ -\\left(n+m+\\frac12\\right)\\log\\left( 1+\\frac 3N \\left(n+m-\\frac N3\\right) \\right) $$\n$$ -\\left(n-m+\\frac12\\right) \\log\\left( 1+\\frac 3N\\left(n-m-\\frac N3\\right) \\right)$$\n$$ -\\left(N-2n+\\frac12 \\right)\\log\\left(1+\\frac 3N\\left(\\frac{2N}{3}-2n\\right)\\right)\\Bigg\\}.$$\n\n\\noindent\nNext we expand the logarithms in the exponent. For the first order terms we obtain\n$$-\\frac 3N \\bigg[\\left(n+m+\\frac 12\\right)\\left(n+m-\\frac N3\\right)+\\left(n-m+\\frac12\\right)\\left(n-m-\\frac N3\\right)+ $$\n$$\\left(N-2n+\\frac 12\\right)\\left(\\frac{2N}{3}-2n\\right)\\bigg] $$ \n$$= -\\frac 3N \\left[\\left(n+m-\\frac N3\\right )^2 + \\left(n-m-\\frac N3 \\right)^2+\\left( \\frac{2N}{3}-2n\\right)^2\\right] $$\n$$= -\\frac{3}{N}\\left( 6\\,\\tilde{n}^2+2m^2\\right),$$\nwhere $ \\tilde{n}:= \\left(n-\\frac N3\\right)$.\\\\\n\n\\noindent\nThe second order terms yield\n$$ \\frac 12 \\left(\\frac 3N\\right)^2\\bigg[ \\left(n+m+\\frac 12\\right)\\left(n+m-\\frac N3\\right)^2+\\left(n-m+\\frac 12\\right)\\left(n-m-\\frac N3\\right)^2$$\n$$+\\left(N-2n+\\frac12\\right)\\left( \\frac{2N}{3}-2n\\right)^2 \\bigg] $$\n$$= \\frac12 \\left(\\frac 3N \\right)^2 \\left[ -6\\tilde{n}^3+(2N+3)\\tilde{n}^2+6\\tilde{n}m^2 +\\left( \\frac{2N}{3}+1 \\right)m^2 \\right] $$\n$$= \\frac 3N \\left( 3\\tilde{n}^2+m^2 \\right) + \\frac{27}{N^{2}}\\left(-\\tilde{n}^3+\\tilde{n}m^2 \\right) +O\\left( \\frac{ \\tilde{n}^2+m^2}{ N^{2}}\\right).$$\n\\noindent\nThe third order terms contribute\n$$-\\frac 13\\left(\\frac 3N \\right)^3 \\bigg[\\left(n+m+\\frac12\\right)\\left(n+m-\\frac N3\\right)^3+\\left(n-m+\\frac12\\right)\\left(n-m-\\frac N3\\right)^3$$\n$$+\\left(N-2n+\\frac12\\right)\\left(\\frac{2N}{3}-2n\\right)^3\\bigg]$$\n$$= \\frac{18(\\tilde{n}^3-\\tilde{n}m^2)}{N^{2}}+O\\left(\\frac{\\tilde{n}^4+m^4}{N^{3}}\\right) $$ \n\\noindent\nAs $d=2m$ and $t=2n$ satisfy $(2.1)$, the contribution of the remaining terms is dominated by that of the fourth order terms and equals\n$$O\\left(\\frac{\\tilde{n}^4+m^4}{N^{3}}\\right). $$\n\nCollecting the results of these computations we arrive at \\smallskip\n\n\\noindent\n\\begin{equation} \n \\begin{split}\n & \\qquad \\qquad \\qquad P(D_N=2m,T_N =2n) = \\frac{3^{\\frac 32}}{ 2\\pi N}\\,\\\\ \n &\\times \\exp\\bigg\\{ -\\frac{3(3\\tilde{n}^2+m^2)}{N}-\\frac{9(\\tilde{n}^3 \n -\\tilde{n}m^2)}{N^2} +O\\left(\\frac1N+ \\frac{\\tilde{n}^4+m^4}{N^3}\\right) \\bigg\\} , \n\\end{split} \n\\end{equation}\n\n\\noindent\nprovided $m\\leq n$ are integers between $1$ and $\\frac12 N$ satisfying $m<\\sqrt{2N\\log N}$ and \n$|\\tilde n|=\\left|n-\\frac N3\\right|<\\sqrt{N\\log N}$. However, we shall also use $(2.2)$ if these inequalities do not hold, since in that case both left- and right-hand members of $(2.2)$ are negligible for our purposes.\n\n \nBy Taylor expansion of the integrand about $x=m$, we find that for integer $0\\frac12 $ we write $r=m+\\theta$ where $m=\\lfloor r\\rfloor$ and $\\theta= \\mbox{frac} (r)=r-\\lfloor r \\rfloor \\in[0,1)$ denote the integer and fractional parts of r respectively. Then for $r<\\sqrt{2N\\log N}$ and $|\\tilde n| = \\left|n-\\frac N3\\right|<\\sqrt{N\\log N}$,\n\\begin{eqnarray*}\n P(2\\leq D_N\\leq 2r,T_N =2n) = P(2\\leq D_N \\leq 2m,T_N =2n)= \\qquad \\\\[3mm] \n \\frac{3^{\\frac32}}{ 2\\pi N}e^{-\\left\\{\\frac{9\\tilde{n}^2}{N}+ \\frac{9\\tilde{n}^3}{N^2}+ O\\left(\\frac1N+\\frac{\\tilde{n}^4}{N^3}\\right)\\right\\}} \\bigg[ \\int\\limits_{[\\frac12,r)}e^{\\frac{-3x^2}{N}+\\frac{9\\tilde{n}x^2}{N^2}}dx \n+ \\int\\limits_{[r,m+\\frac12)} e^{\\frac{-3x^2}{N}+\\frac{9\\tilde{n}x^2}{N^2}} dx \\bigg] .\n\\end{eqnarray*}\n\\noindent\nEvaluating the second integral by expanding the integrand about the point $x=m+\\frac12 $, we arrive at\n\\begin{eqnarray*}\n&& P(2\\leq D_N\\leq 2r,T_N =2n) = \n\\frac{3^{\\frac32}}{ 2\\pi N}e^{-\\left\\{ \\frac{9}{N}\\tilde{n}^2 +\\frac{9}{N^2}\\tilde{n}^3+\n O\\left(\\frac1N+\\frac{1}{N^3}\\tilde{n}^4 \\right)\\right\\}} \\\\ && \\quad \\quad\\times \n \\left[ \\int_{[\\frac12,r)} e^{-\\frac{3}{N}x^2 +\\frac{9}{N^2}\\tilde{n}x^2}dx -\n e^{-\\frac{3}{N}r^2 +\\frac{9}{N^2}\\tilde{n}r^2 } \\left(\\mbox{frac} (r)-\\frac12+O\\left(\\frac rN\\right)\\right) \\right]. \n\\end{eqnarray*}\n\\noindent\nAgain we may use this for all $r>0$ and integer $n\\leq\\frac12 N$.\n\nChoose $w>0$ and $r=w\\sqrt{\\frac n2}$. We have\n\\begin{eqnarray*}\n&& P(00$,\n\\begin{equation}\n \\begin{split}\n P\\left( 00$. Since it is identical to $(1.6)$, this proves the third statement of the theorem.\n\n\nIt remains to prove that any closed interval of length $O(N^{-1})$ that does not contain the origin has probability $O(N^{-1})$ under $\\Psi_N$ and hence $F_N$. Clearly, this will imply the second statement of the theorem. Obviously, the only term in $(2.8)$ that we need to consider is \n\\begin{equation*}\nR(w) = \\frac{\\Lambda_N(w)}{\\sqrt{N}}\n\\end{equation*}\n\\begin{equation}\n= -\\sqrt{\\frac{3}{2N}} \\varphi (w) \\sum_{0\\leq n\\leq N} \\frac{3}{\\sqrt{\\pi N} }e^{-\\frac 9N\\left(n - \\frac N3\\right)^2} \n \\left(\\mbox{frac} \\left(w\\sqrt{2n} \\right)-\\frac12 \\right) ,\n\\end{equation}\nas the remainder of the expansion obviously has bounded derivative. \n\n We begin by noting that if for a given $w>0$, $w\\sqrt{ 2n}$ is an integer for some $1\\leq n\\leq N$, then $\\mbox{frac} \\left(w\\sqrt{ 2n}\\right)$ and hence $R$ has a jump discontinuity at this value of $w$. In the range where\n$|n- \\frac N3|= x\\sqrt{N}$ for $|x|\\leq y$, there can be a most $wy$ such integer values of $n$. To see this, simply note that if $w\\sqrt{ 2n}=k$ and $w\\sqrt{2n'}=k+1$ , then $|n'-n| \\geq \\frac{ 2\\sqrt{N}}{w}$ , so there can be only \n$\\frac{2y\\sqrt{N}}{ \\frac{2\\sqrt{N}}{w} }=wy$ values of $n$ in the required interval. Such a value of n contributes an amount $O\\left(N^{-1}\\varphi(w)e^{-9x^2}\\right)$ to the jump discontinuity at $w$, and hence $R(w)-R(w-0)= O(N^{-1})$ at such a point $w$. Incidentally, this proves the second part of Theorem 1.1.\n\n Choose $\\epsilon>0$ and consider two such jump points $w\\not=w'$ in $[\\epsilon , \\infty)$ with $w\\sqrt{ 2n}=k$ and $w'\\sqrt{2n'}=k'$ for integers $k, k', n$ and $n'$ with $(n-\\frac N3)=x\\sqrt{N},\\ \\left(n'-\\frac N3\\right)=x'\\sqrt{N} $ and $|x|\\vee | x'|\\leq y$. Suppose that $(w'-w)=O(N^{-1})$ and hence $\\frac{w'-w}{w}=O(N^{-1})$ since $w\\geq \\epsilon$. For given $w$, $n$ and $k$, we ask how many integer values of $n'$ satisfy these conditions.\n\nFirst we note that, for some positive $c$ there are only at most $cw(y+1)$ possible choices for $k'$ since $\\sqrt{2n} = \\sqrt{2\\frac N3 +2x\\sqrt{N}} = \\sqrt{\\frac{2N}{3}} +\\sqrt{\\frac32}x+O\\left( \\frac{y^2}{\\sqrt{N}} \\right), \\sqrt{2n'} =\\sqrt{\\frac23 N} +\\sqrt{\\frac 32}x'+O\\left(\\frac{y^2}{\\sqrt{N}}\\right)$ and hence $|k'-k|\\leq 2wy + O\\left( w\\frac{y^2}{\\sqrt N}+|w'-w|\\sqrt{N}\\right)\\leq \\left(\\frac c2 \\right)w(y+1)$. For each choice of $k'$, the corresponding $n'$ satisfies $n' =\\frac12 \\left(\\frac{k'}{w'}\\right)^2$ for some admissible $w'$, and since $w,w'\\geq \\epsilon$ and $(w'-w)=O(N^{-1})$, this leaves a range of order $O\\left( \\left(\\frac{k'}{w'}\\right)^2 N^{-1} \\right)=O(1)$ for $n'$. Hence, for some $C>0$, there are at most $Cw(y+1)$ possible values of $n'$ for which there exists an integer $k'$ with $(w'-w) = O(N^{-1})$. By the same argument as above, the total contribution of discontinuities to$|R(w')-R(w) |$ is $O(N^{-1})$ as long as $|w-w'|= O(N^{-1})$. As any closed interval of length $O(N^{-1})$ that does not contain the origin is bounded away from $0$, this holds for the sum of the discontinuities in such an interval.\n\n At all other points $w>0$, $R$ is differentiable and the derivative of $\\mbox{frac} \\left(w\\sqrt{ 2n}\\right)$ equals $\\sqrt{2 n}$. Hence the derivative of $R$ is $O(1)$ and its differentiable part contributes at most $O(N^{-1})$ to the probability of any interval of length $O(N^{-1})$. This completes the proof of the Theorem 1.1.\n\n\n\n\\section{Evaluation of the oscillatory term}\n\nLet $W$ denotes a r.v. with non negative c.f. $\\psi(t)\\ge 0$ of\nsupport contained in $[-1,1]$ and exponential decay of density \nof type $\\exp\\{-|x|^{2\/3}\\},\\,x\\to\\infty$, \n\\citep[see e.g.] [p. 85] {Bhattacharya-Rao:1986\/2}.\nIntroduce r.v. $w_N := w +N^{-1\/2}(\\log N)^{-1}\\4 W, \\, w >0$ and \nlet $c>0$ denote an positive absolute constant. \nThen we may bound the normal approximation\nerror in $(1.6)$ using similar arguments as in the proof of \nthe well-known smoothing inequality, (see Lemma 12.1 of Bhattacharya and Rao),\n obtaining, for $w\\ge 1$,\n\\begin{equation}\n\\begin{split}\n\\,N^{-1\/2}\\bigl|\\mathbf E \\Lambda_N(w_N)\\bigr| \\le & \n\\, \\bigl|\\mathbf E \\bigl(F_N(w_N) -\\Phi(w_N)\\bigr)\\bigr| +\ncN^{-1}\\\\ \n& \\,\\le \\sup_{x\\in[w-1\/2,w+1\/2]}\\bigl|F_N(x) -\\Phi(x)\\bigr|+cN^{-1}, \n\\end{split}\n\\end{equation} \nwhere\n$$\\Lambda_N(w):= \\, -\\varphi(w)\\4 \n\\sum_{1\\le n\\le N}\\frac {3^{3\/2}}{\\sqrt{2 \\pi N}}\n\\exp\\{-\\frac 9 N(n - \\frac N 3)^2\\}(\\mbox{frac}(w \\sqrt {2n})- 1\/2 ).$$\nWe start with the following Fourier series expansion\n\\begin{eqnarray} \n\\tau(x):= frac(x)-1\/2 = -\\sum_{k=1}^{\\infty} 2\\4 \\frac{ sin(2 \\pi\\4 k\\4 x)}\n{2\\4 k\\4 \\pi}, \n\\end{eqnarray*}\nwhich holds for all nonintegral $x$. \n\nNote that by the properties of $W$ (i.e. the vanishing of Fourier coefficients)\n$$ \n\\mathbf E \\tau(w_N\\sqrt{2n})= -\\sum_{k=1}^{M_n} \\mathbf E \\frac{ sin(\n 2\\pi\\4 k\\4 \\sqrt{2n}(w+N^{-1\/2}(\\log N)^{-1}\\4 W))} { k\\4 \\pi},\n$$\nwhere $M_n:=[\\sqrt N\\log N\/(2\\pi\\sqrt {2n})]+1$, i.e. $M_n = O(\\log N)$ for \n$|n-N\/3|<\\sqrt{N\\log N}$.\n\nRewriting $\\Lambda_N(w)$ in $(3.1)$ in the form\n\\begin{equation}\n\\Lambda_N(w) := \\, -\\frac {3^{3\/2}} {(2 \\4 \\pi\\4 N)^{1\/2}} \\varphi(w) \n\\sum_{n=1}^{N} exp\\{-9\\4 (\\tilde{n}^2\/N\\}\\4 \n\\tau\\bigl(w\\4(2 \\4 n)^{1\/2}\\bigr),\n\\end{equation}\nwhere $\\tilde{n}:=n-N\/3$,\nwe get \n\\begin{equation}\n\\begin{split}\n\\mathbf E \\Lambda_N(w_N) =& \\, \\sqrt{\\frac 3 2}\\pi^{-1} \n\\sum_{k=1}^{M} \\frac 1 k \n\\lambda_{N,k}+O(N^{-3}), \\,\\, \n\\text{where}\\,\\, M:=[\\log N]\\,\\, \\text{and}\\\\\n\\lambda_{N,k} :=& \\, \\frac 3{\\sqrt{\\pi \\4 N}} \\mathbf E \\varphi(w_N)\\sum_{n=1}^{N} \nexp\\{-9\\4 \n\\tilde{n}^2\/N\\}\\4 sin(2\\pi\\4 k\\4 w_N\\4 \\sqrt{2n}).\n\\end{split}\n\\end{equation}\nIn the arguments of the $\\sin$ function we\n use a Taylor expansion, for $|n-N\/3|<\\sqrt{N\\log N}$, \n$$\n\\sqrt{n} = \\sqrt{N\/3} + \\sqrt 3 \\tilde{n}\/(2\\sqrt N)+\nO\\bigl(\\tilde{n}^2\/N^{3\/2} \\bigr).\n$$\nThus, for $|\\tilde{n}|<\\sqrt{N\\log N}$, \n\\begin{equation}\n sin(2\\pi\\4 k\\4 w_N\\4 \\sqrt{2n})= sin\\bigl(d_0 + \\4 \\pi\\4 d_1 \\tilde{n} \\bigr) \n+ O\\bigl(k\\4 w_N N^{-3\/2}\\4 \\tilde{n}^2\\bigr),\n\\end{equation}\nwhere $ d_0 :=2\\pi\\4k \\4 w_N\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}$, \n$d_1 := k\\4 \\4 w_N\\4\n(\\frac 3 2)^{1\/2}\\4\/ \\sqrt N$. %\nHence we may write\n\\begin{equation}\n\\begin{split}\n \\lambda_{N,k} =& \\frac 3 {\\sqrt{\\pi \\4 N}} \\mathbf E \\varphi(w_N) \n\\sum_{n\\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} exp\\{-9\\4 \n\\tilde{n}^2\/N\\}\\4\n\\sin\\bigl(d_0 + \\4 2\\pi \\4d_1 \\4\\tilde{n}\\bigr)\\\\\n&\\quad + O(k N^{-1\/2} \\4 \\log N ). \n\\end{split}\n\\end{equation}\nWe shall now evaluate the theta sum on the left hand side using\nPoisson's formula, \n\\citep[see e.g.][p. 189]{Mumford:1983}.\n\\begin{equation}\n\\sum_{m \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \\exp\\{-z\\4 m^2 +i2\\pi\\4 m \\4 b\\} = \\pi^{1\/2}z^{-1\/2}\n\\sum_{l \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }}\n\\exp\\{- \\pi^2 \\4 z^{-1}(l- b)^2\\},\n\\end{equation}\nwhere $b \\in \\mathbb R$, $\\Re z >0$ and $z^{1\/2}$ denotes the branch with \npositive real part.\nWriting $sin(x)=(\\exp[i\\4 x] - \\exp[-i\\4 x])\/2$ in (3.6) \nand assuming for simplicity $N\/3 \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }$ we may replace summation over n by\nsummation over $ m:=\\tilde{n}= n - N\/3 \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }$ in (3.6). \nApplying now (3.7) we have to bound the imaginary part of \n expectations of theta functions of type \n\\begin{eqnarray}\nI_k:=\\frac 3 {\\sqrt{\\pi\\4 N}} \\mathbf E \\varphi(w_N)\\exp\\{\\4 i\\4 d_0 \\}\n\\sum_{m\\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \\exp\\{-\n9\\4 m^2 \\4 N^{-1} + i\\4 2\\pi\\4 d_1\\4m\\}.\n\\end{eqnarray*}\nWe obtain for $ k \\le M =[\\log N]$ that $|d_1| \\le 2\n \\4N^{-1\/2}(\\log N) |w_N| \\le 4 \\4N^{-1\/2}(\\log N)^2 $ with probability\n $1- O(N^{-3\/2})$ by the assumption $w \\le \\log N$.\n Hence the dominant term\nin (3.9) below is the term with $l=0$ and we obtain with \n$c_{N,k}:= \\exp\\{ \\4 2\\pi\\4 i\\4 \\4k \\4 w_N\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}\\}$ \n\\begin{equation}\n\\begin{split}\nI_k =& \\, \\mathbf E c_{N,k}\\varphi(w_N)\\4 \\sum_{l \\in \\hbox{\\rm Z\\negthinspace \\negthinspace Z }} \n\\exp\\{ - N \\4( l- d_1)^2\\4 \\pi^2\/9\\}\n\\\\\n = & \\, \\mathbf E c_{N,k}\\4 \\varphi(w_N)\\exp\\{- N\\4 d_1^2\\4 \\pi^2\/9\\} + O\\bigl(N^{-3\/2}\\bigr)\n\\\\ \n =& \\,\\4 f_{N,k}\\4 \\varphi(w) \\exp\\{- \\pi^2 k^2 \\4 w^2\/6 \n+ \\4 i\\4 \\42\\pi k \\4 w\\4 (\\frac 2 3)^{1\/2} \\sqrt{N}\\}\n+ O\\bigl(N^{-1\/2}(\\log N)^4\\bigr),\n\\end{split}\n\\end{equation}\nwhere $f_{N,k}:= \\psi\\bigl(2\\pi\\4 (\\frac 2 3)^{1\/2} \\4 \\frac k {\\log\n N}\\bigr)= 1+ O\\bigl((k\/\\log N)^2\\bigr)$. \nUsing the equation (3.9) in (3.4) we get\n\\begin{equation}\n\\begin{split}\n\\mathbf E \\Lambda_N(w_N) =& \\, \\sqrt\\frac 3 2 \n\\varphi(w)\\Im \\sum_{k=1}^M \\frac {f_{N,k}} {k\\pi}\n\\exp\\bigl\\{- \\frac{\\pi^2}6\\4 k^2\\4 w^2\\4+ \n2\\pi\\4 i\\4 \\4k \\4 w\\4 \\sqrt{\\frac{2N}3}\\bigr\\}\\\\\n& \\, \\quad + O\\bigl(N^{-1\/2}(\\log N)^5\\bigr).\n\\end{split}\n\\end{equation}\n\nHence, there exists a constant $c_0(w) >0$ such that\n\\begin{equation}\n|\\mathbf E \\Lambda_N(w_N)| > c_0(w) >0,\n \\end{equation} \nprovided that $4\\4 w\\4 \\sqrt{\\frac{2 N} 3} $ is an odd integer, which\nproves the assertion $(1.10)$. \n\n\\renewcommand\\bibsection{\\section*{REFERENCES}}\n\\bibliographystyle{ims}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA hypergraph $H = (V,E)$ is said to be bipartite or 2-colorable\nif the vertex set $V$ can be partitioned into two disjoint sets $V_1$ and $V_2$\nsuch that every edge $e\\in E$ has non-empty intersections with both the partitions.\nIn the case of graphs, one can easily find the two partitions from any given instance of\n$H$ by breadth first search.\nHowever, the problem turns out to be notoriously hard if edges of size more than 2 are present.\nIn fact, in the case of bipartite 3-uniform and 4-uniform hypergraphs,\nit is well known that the problem is NP-hard~\\cite{Dinur_2005_jour_Combinatorica,Khot_2014_conf_SODA}.\n\nIn general, finding a proper 2-coloring is relatively easy if the hypergraph is sparse. \nIn an answer to a question asked by Erd\\\"os~\\cite{Erdos_1963_jour_NordikMat} on 2-colorability of uniform hypergraphs, it is now known that for large $m$,\nany $m$-uniform hypergraph on $n$ vertices with at most \n$2^m0.7\\displaystyle\\sqrt{\\frac{m}{\\ln m}}$ edges is 2-colorable~\\cite{Radhakrishnan_1998_conf_FOCS}. As pointed in~\\cite{Radhakrishnan_1998_conf_FOCS}, the result can also be extended to \nnon-uniform hypergraphs with minimum edge size $m$. \nHowever, it is much worse if the restriction on the minimum edge size and the\nnumber of hyperedges is not imposed. Even when a hypergraph is 2-colorable, the best\nknown algorithms~\\cite{Alon_1996_jour_NordicJComput,Chen_1996_conf_IPCO}\nrequire $O\\left((n\\ln n)^{1-1\/M}\\right)$ colors to properly color the hypergraph\nin polynomial time,\nwhere $M$ is the maximum edge size, also called dimension, of the hypegraph.\nIn recent years, 2-colorability of random hypergraphs has also received considerable attention.\nThrough a series of works~\\cite{Achlioptas_2008_conf_FOCS,CojaOghlan_2012_conf_SODA,Panangiotou_2012_conf_STOC},\nit is now established that random uniform hypergraphs are 2-colorable only when\nthe number of edges are at most $Cn$, for some constant $C>0$.\nThus, it is evident that coloring relatively dense hypergraphs is difficult unless the \nhypergraph admits a ``nice\" structure.\n\nIn spite of the hardness of the problem,\nthere are a number of applications that require hypergraph coloring algorithms.\nFor instance, such algorithms have been used for approximate DNF counting~\\cite{Lu_2004_jour_SIAMJDiscMath}, as well as in various resource allocation and scheduling\nproblems~\\cite{Capitanio_1995_jour_IJPP,Ahuja_2002_conf_APPROX}.\n The connection between ``Not-All-Equal\" (NAE) SAT\nand hypergraph 2-coloring also demonstrate its significance in context of satisfiability problems. \nAmong the various approaches studied in the literature, perhaps\nthe only known non-probabilistic instances of efficient 2-coloring are in the cases \nwhere the hypergraph is $\\alpha$-dense, 3-uniform and bipartite~\\cite{Chen_1996_conf_IPCO}, \nor where the hypergraph is $m$-uniform and its every edge has equal number of vertices of either colors~\\cite{McDiarmid_1993_jour_CombProbComput}.\n\nIn this paper, we consider the problem of coloring random non-uniform hypergraphs of dimension $M$,\nthat has an underlying planted bipartite structure. We present a polynomial time algorithm\nthat can properly 2-color instances of the random hypergraph with high probability whenever\nthe expected number of edges in at least $dn\\ln n$ for some constant $d>0$.\nTo the best of our knowledge, such a model has been only considered \nby Chen and Frieze~\\cite{Chen_1996_conf_IPCO}, who extended a graph coloring \napproach of Alon and Kahale~\\cite{Alon_1997_jour_SIAMJComput} to \npresent an algorithm for \n 2-coloring of 3-uniform bipartite hypergraphs with $dn$ number of edges.\nTo this end, our work generalizes the results of \\cite{Chen_1996_conf_IPCO} to\nnon-uniform hypergraphs, and it is the first algorithm that is guaranteed to properly color\nnon-uniform bipartite hypergraphs using only two colors. We also discuss the possible extension \nof our approach to the case of non-uniform $k$-colorable hypergraphs.\n\n\\subsection*{The Main Result}\nBefore stating the main result of this paper, we present the planted model under \nconsideration, which \nis based on the model that is studied in~\\cite{Ghoshdastidar_2015_arxiv}.\nThe random hypergraph $H_{n,(p_m)_{m=2,\\ldots,M}}$ is generated\non the set of vertices $V = \\{1,2,\\ldots,2n\\}$, which is arbitrarily split into \ntwo sets, each of size $n$, and the sets are colored with two different colors.\nGiven a integer $M$, and $p_2,\\ldots,p_M\\in[0,1]$, the edges of the hypergraph\nare randomly added in the following way. All the edges \nof size at most $M$ are added independently, and for any $e\\subset V$, \n\\begin{align*}\n \\P(e\\in E) = \\left\\{ \\begin{array}{ll}\n p_m & \\text{if } e \\text{ is not monochromatic and } {|e|=m}, \\\\\n 0\t & \\text{otherwise}.\\\\\n \\end{array}\\right.\n\\end{align*}\nWe prove the following result.\n\\begin{theorem}\n\\label{thm_spec_color}\n Assume $M=O(1)$. There is a constant $d>0$ such that if \n \\begin{equation}\n \\sum\\limits_{m=2}^M p_m \\binom{2n}{m} \\geq {dn\\ln n}, \n \\end{equation}\n then with probability \n $(1-o(1))$, Algorithm~\\ref{alg} (presented in next section) finds a proper 2-coloring of the random non-uniform bipartite hypergraph $H_{n,(p_m)_{m=2,\\ldots,M}}$. \n\\end{theorem}\nIt is easy to see that the expected number of edges in the hypergraph is \n$\\Theta\\left(\\sum_{m=2}^M p_m\\binom{2n}{m}\\right)$, and so the condition may be stated\nin terms of expected number of edges.\n\n\\subsection*{Organization of this paper}\nThe rest of the paper is organized in the following manner.\nIn Section~\\ref{sec_algorithm}, we present our coloring algorithm, followed by a proof of \nTheorem~\\ref{thm_spec_color} in Section~\\ref{sec_proof}. In the concluding remarks in \nSection~\\ref{sec_conclusion}, we provide discussions about the key assumptions made in this work,\nand also the possible extensions of our results to $k$-coloring and strong coloring of non-uniform\nhypergraphs. The appendix contains proofs of the lemmas mentioned in Section~\\ref{sec_proof}.\n\n\\section{Spectral algorithm for hypergraph coloring}\n\\label{sec_algorithm}\n\nThe coloring algorithm, presented below, is similar \nin spirit to the spectral methods of~\\cite{Alon_1997_jour_SIAMJComput,Chen_1996_conf_IPCO},\nbut certain key differences exist, which are essential to deal with\nnon-uniform hypergraphs. \n\nGiven a hypergraph $H = (V,E)$, \nan initial guess of the color classes is formed by exploiting the spectral properties of a certain matrix\n$A\\in\\mathbb{R}^{|V|\\times|V|}$ defined as \n\\begin{align}\n A_{ij} = \\left\\{ \\begin{array}{rl}\n \\displaystyle\\sum_{e\\in E: e\\ni i,j} \\frac{1}{|e|}\t& \\text{if } i\\neq j, \\text{ and} \\\\\n \\displaystyle\\sum_{e\\in E: e\\ni i} \\frac{1}{|e|}\t& \\text{if } i= j. \n \\end{array}\\right.\n \\label{eq_defnA}\n\\end{align}\nThe above matrix has been used in the literature to construct the Laplacian of a \nhypergraph~\\cite{Bolla_1993_jour_DiscreteMath,Ghoshdastidar_2015_arxiv},\nand is also known to be related to the affinity matrix of the star expansion of \nhypergraph~\\cite{Agarwal_2006_conf_ICML}. \nThe use of matrix $A$ is in contrast to the adjacency based graph construction of~\\cite{Chen_1996_conf_IPCO} that is likely to\nresult in a complete graph if the hypergraph is dense.\n\nThe later stage of the algorithm considers an iterative procedure that\nis similar \nto~\\cite{Alon_1997_jour_SIAMJComput,Chen_1996_conf_IPCO}, but uses a \nweighted summation of neighbors. Such weighting is crucial while\ndealing with the edges of\ndifferent sizes.\n\n\\begin{varalgorithm}{COLOR}\n\\caption {-- Colors a non-uniform hypergraph $H$:}\n\\label{alg}\n\\begin{algorithmic}[1]\n \\STATE Define the matrix $A$ as in~\\eqref{eq_defnA}.\n \\STATE Compute\n $x^A = \\underset{\\Vert x \\Vert_2 = 1}{\\textup{arg min~}} x^TAx$.\n \\STATE Let $T = \\lceil \\log_2 n\\rceil$, $V_1^{(0)} = \\{ i\\in V: x_i^A \\geq 0\\}$ and \n $V_2^{(0)} = \\{ i\\in V: x_i^A < 0\\}$.\n \\FOR {$t = 1,2,\\ldots, T$}\n \\STATE Let \n $V_1^{(t)} = \\left\\{ i\\in V: \\sum\\limits_{j\\in V_1^{(t-1)}\\backslash\\{i\\}} A_{ij} <\n \\sum\\limits_{j\\in V_2^{(t-1)}\\backslash\\{i\\}} A_{ij} \\right\\}$, \n \\newline and $V_2^{(t)} = V\\backslash V_1^{(t)}$.\n \\ENDFOR\n \\IF {{$\\exists e\\in E$} such that $e\\subset V_1^{(T)}$ or $e\\subset V_2^{(T)}$}\n \\STATE Algorithm FAILS.\n \\ELSE\n \\STATE 2-Color $V$ according to the partitions \n $V_1^{(T)},V_2^{(T)}$.\n \\ENDIF\n\\end{algorithmic}\n\\end{varalgorithm}\n\n\\section{Proof of Main Result}\n\\label{sec_proof}\n\nWe now prove Theorem~\\ref{thm_spec_color}.\nWithout loss of generality, assume that the true color classes in $V$ are $\\{1,2,\\ldots,n\\}$ and $\\{n+1,\\ldots,2n\\}$.\nAlso, let $W^{(t)}$, $t=0,1,\\ldots,T$, denote the incorrectly colored\nvertices after iteration $t$,\nwith $W^{(0)}$ being the incorrectly colored nodes after initial spectral step.\nWe prove Theorem~\\ref{thm_spec_color} by showing with probability $(1-o(1))$, \nthe size of $W^{(T)} <1$, which implies that all nodes are correctly colored, and hence, the hypergraph must be \nproperly colored.\n\nThe first lemma bounds the size of $W^{(0)}$, \\textit{i.e., } the error incurred at the initial spectral step.\n\\begin{lemma}\n\\label{lem_spectral}\nWith probability $(1-o(1))$,\n$|W^{(0)}| \\leq \\displaystyle\\frac{n}{M^22^{2M+4}}$.\n\\end{lemma}\nNext, we analyze the iterative stage of the algorithm to make the following claim,\nwhich characterizes the vertices that are correctly colored after iteration $t$.\n\\begin{lemma}\n\\label{lem_iteration_charac}\n Let $\\eta = \\displaystyle\\frac{1}{2^{M+2}}\\sum\\limits_{m=2}^M\\frac{p_m(n-1)}{m}\\binom{n-2}{m-2}$. \n For any $t\\in\\{1,\\ldots,T\\}$,\n if $\\sum\\limits_{j\\in W^{(t-1)}\\backslash\\{i\\}} A_{ij} < \\eta$ for any $i\\in V$, then \n $P(i\\in W^{(t)})\\leq n^{-\\Omega(d)}$.\n\\end{lemma}\nNote that there are only $T=\\lceil \\log_2 n\\rceil$ iterations, and $|V| =2n$. \nCombining the result of Lemma~\\ref{lem_iteration_charac} with union bound, we can conclude\nthat with probability $(1-o(1))$, for all iterations $t=1,2,\\ldots,T$, \nthere does not exist any $i\\in V$ such that\n$\\sum\\limits_{j\\in W^{(t-1)}\\backslash\\{i\\}} A_{ij} < \\eta$.\nWe also make the following observation, where $\\eta$ is defined in Lemma~\\ref{lem_iteration_charac}.\n\\begin{lemma}\n\\label{lem_iteration_size}\nWith probability $(1-o(1))$, there does not exist $C_1,C_2\\subset V$ such that $|C_1|\\leq\\frac{n}{M^22^{2M+4}}$,\n$|C_2| = \\frac12 |C_1|$ and for all $i\\in C_2$, $\\sum\\limits_{j\\in C_1\\backslash\\{i\\}} A_{ij} \\geq \\eta$.\n\\end{lemma}\nWe now use the above lemmas to proceed with the proof of Theorem~\\ref{thm_spec_color}.\nLemma~\\ref{lem_spectral} shows that $|W^{(0)}|\\leq \\frac{n}{M^22^{2M+4}}$ with probability $(1-o(1))$.\nConditioned on this event, and due to the conclusion of Lemma~\\ref{lem_iteration_charac},\none can argue that Lemma~\\ref{lem_iteration_size} is violated unless\n$|W^{(t)}| < \\frac12 |W^{(t-1)}|$ for all iteration $t$ with probability $(1-o(1))$. \nThus, in each iteration,\nthe number of incorrectly colored vertices are reduced by at least half. Hence, after \n$T=\\lceil \\log_2 n\\rceil$ iterations, $|W^{(T)}| <1$, which implies that all vertices are correctly colored.\n\n\\section{Discussions and Concluding remarks}\n\\label{sec_conclusion}\nIn this paper, we showed that a random non-uniform bipartite hypergraph of dimension $M$ \nwith balanced partitions can be properly 2-colored with \nprobability $(1-o(1))$ by a polynomial time algorithm.\nThe proposed method uses a spectral approach to form initial guess of the color classes,\nwhich is further refined iteratively.\nTo the best of our knowledge, this is the first work on 2-coloring bipartite non-uniform hypergraphs.\nPrevious works~\\cite{Chen_1996_conf_IPCO,Krivelvich_2003_jour_JAlgo} \nhave only restricted to the case of uniform hypergraphs.\n\n\\subsection*{A note on the assumptions in Theorem~\\ref{thm_spec_color}}\n\nThe key assumptions made in this paper are the following:\n\\begin{enumerate}\n\\item $M = O(1)$, and \n\\item $p_2,\\ldots,p_M$ are such that\nthe expected number of edges is larger than $dn\\ln n$, where $d>0$ is a large constant.\n\\end{enumerate}\nThe assumption $M = O(1)$ is crucial, particularly in Lemma~\\ref{lem_spectral},\nand helps to ensure that $d$ can be chosen to be a constant. This can be avoided \nif $d$ is allowed to increase with $n$ appropriately. We note that a previous \nwork on spectral hypergraph partitioning~\\cite{Ghoshdastidar_2015_arxiv} allows\n$M$ to grow with $n$, but imposes an additional restriction so that the number of \nedges of larger size decay rapidly.\n\nThe second assumption is stronger than the one in \\cite{Chen_1996_conf_IPCO},\nwhere it was shown that a random bipartite 3-uniform hypergraph can be properly\n2-colored with high probability if the expected number of edges is $dn$.\nThis is due to the use of matrix Bernstein inequality~\\cite{Tropp_2012_jour_FOCM}\nin Lemma~\\ref{lem_spectral} that does not provide useful bounds in the most sparse \ncase. On the other hand, Chen and Frieze~\\cite{Chen_1996_conf_IPCO}\nuse the techniques of Kahn and Szemeredi~\\cite{Friedman_1989_conf_STOC}\nthat allows them to work in the most sparse regime. \nHowever, it is not clear how the \nsame techniques can be extended even to uniform hypergraphs of higher order.\nThus, it remains an open problem whether a similar result can be proved when the number of edges in the hypergraph grows linearly with $n$.\n\n\\subsection*{$k$-coloring of hypergraphs}\nThough Algorithm~\\ref{alg} has been presented only for the hypergraph 2-coloring problem,\none may easily extend the approach to achieve a $k$-coloring,\nwhere the objective is to color the vertices of the hypergraph with $k$ colors such that no edge \nis monochromatic.\nA possible extension of Algorithm~\\ref{alg} is as follows:\n\\begin{enumerate}\n \\item\n In Step~2, compute the eigenvectors corresponding to the $(k-1)$ smallest eigenvalues of $A$. \n \\item\n Use $k$-means algorithm~\\cite{Ostrovsky_2013_jour_JACM} to cluster rows of the eigenvector matrix into $k$ groups,\n and define the initial guess for the color classes $V_1^{(0)},\\ldots,V_k^{(0)}$ in Step~3 according\n to the above clustering.\n \\item\n The iterative computation in Step~6 is modified by defining\n \\begin{displaymath}\n \\qquad\n V_l^{(t)} = \\left\\{ i\\in V: \\sum\\limits_{j\\in V_l^{(t-1)}\\backslash\\{i\\}} A_{ij} <\n \\sum\\limits_{j\\in V_{l'}^{(t-1)}\\backslash\\{i\\}} A_{ij} \\text{ for all } l'\\neq l\\right\\}\n \\end{displaymath}\n for $l=1,2,\\ldots,(k-1)$, and $V_k^{(t)} = V\\backslash \\left(\\bigcup_{l>> \\widehat G(c) \\\\\n@AAA @AAA \\\\\n\\widehat G({\\mathcal O},a) @>>> \\widehat G({\\mathcal O},c).\n\\end{CD}\n\\end{equation*}\nOne can also view $\\widehat G({\\mathcal O},c)$ as the projective limit of\n$\\mathbf G({\\mathcal O})\/\\Gamma({\\mathfrak a})$ over nonzero ideals ${\\mathfrak a}\\subseteq {\\mathcal O}$ and similarly for\n$\\widehat G({\\mathcal O},a)$. Thus they are profinite (and hence compact) groups,\nwhile $\\widehat G(c)$ and $\\widehat G(a)$ are locally compact. It is then\neasy to see that the two horizontal maps are surjective and have the same\nkernel which is called the \\emph{congruence subgroup kernel} $C(S, \\mathbf G)$. \n\nFrom a more general perspective, the congruence subgroup problem is the\ndetermination of $C(S,\\mathbf G)$. The case when $C(S,\\mathbf G)=1$ is equivalent to\nevery $S$-arithmetic subgroup being an $S$-congruence subgroup.\n\n\\subsection{Reductions}\n\nThe congruence subgroup problem admits a number of reductions. The functor\n$\\mathbf G\\to C(S,\\mathbf G)$ satisfies a weak form of exactness outlined in\n\\cite{Ra1}*{Introduction}. Since $C(S,\\mathbf G)=1$ when $\\mathbf G$ is finite or the\nadditive group $\\mathbf G_a$, this implies that $C(S,\\mathbf G) = C(S, \\mathbf G^0\/\\mathbf\nN_{\\mathbf G})$, where $\\mathbf N_{\\mathbf G}$ is the unipotent radical of $\\mathbf G$. We thus\nmay assume that $\\mathbf G$ is connected and reductive. A theorem of Chevalley\n\\cite{Chevalley} based on class field theory implies that $C(S,\\mathbf\nT)=1$ for $\\mathbf T$ a $k$-torus. Together with the weak exactness\nproperty, this implies that $C(S,\\mathbf G) = C(S,\\mathpsscr D\\mathbf G)$ where\n$\\mathpsscr D\\mathbf G$ is the derived group (see also\n\\cite{PlatonovSaromet}). It thus suffices to assume that $\\mathbf G$ is\nconnected and semisimple.\n\nIf $\\mathbf G$ is not simply connected then $C(S,\\mathbf G)$ can be infinite.\nSpecifically let $\\widetilde{\\mathbf G}$ be the simply connected covering group of\n$\\mathbf G$ and let $\\mathbf B = \\Ker (\\widetilde{\\mathbf G}\\to \\mathbf G)$. If all $k$-simple\ncomponents $\\mathbf H$ of $\\mathbf G$ satisfy $k_v\\text{-rank}\\: \\mathbf H > 0$ for some\n$v\\in S$, then $\\Coker(C(S,\\widetilde{\\mathbf G}) \\to C(S,\\mathbf G))$ will contain an\nisomorphic copy of the infinite group $\\mathbf B({\\mathbb A}_{k,S})\/\\mathbf B(k)$,\nwhere ${\\mathbb A}_{k,S}$ denotes the $S$-adeles of $k$\n\\citelist{\\cite{SerreBourbaki} \\cite{Ra1}}. Thus we will make the\nassumption that $\\mathbf G$ is simply connected.\n\nAny simply connected group is a direct product of almost $k$-simple groups,\nso we may assume $\\mathbf G$ is almost $k$-simple. We may then write $\\mathbf G =\\Res_{k'\/k}\n\\mathbf G'$, where $\\mathbf G'$ is an absolutely almost simple group over a finite\nextension $k'$ over $k$. Since $C(S,\\mathbf G) = C(S',\\mathbf G')$ where $S'$ consists\nof all places of $k'$ lying over places of $S$, we may assume that $\\mathbf G$ is\nconnected, simply connected and absolutely almost simple.\n\n\\subsection{Some known results}\n\\label{ssectKnownResults}\n\nThe congruence subgroup kernel has been considered extensively by many\nauthors; see the survey \\cite{PrasadRapinchukSurvey}. In particular, Bass,\nMilnor, and Serre \\cite{BMS} proved that $C(S, \\mathbf G)$ is finite for the\ngroups $\\SL_n$, $n\\ge 3$, and $\\SP_{2n}$, $n\\ge 2$; in fact they prove that\n$C(S,\\mathbf G)$ is trivial unless $k$ is totally imaginary and $S=S_\\infty$ in\nwhich case $C(S,\\mathbf G)\\cong \\mu(k)$, the roots of unity in $k$. Serre\n\\cite{se3} later treated the case $\\SL_2$ and obtained the same\ndetermination of $C(S,\\mathbf G)$ if $|S|\\ge 2$; if $|S|=1$ he proves that\n$C(S,\\mathbf G)$ is infinite.\n\nLet $S\\text{-rank}\\: \\mathbf G = \\sum_{v\\in S} k_v\\text{-rank}\\: \\mathbf G$.\nFor a global field $k$ (that is, a number field or a function field of an\nalgebraic curve over a finite field) Serre \\cite{se3} has conjectured%\n\\footnote{The hypothesis that $k_v\\text{-rank}\\: \\mathbf G>0$ for all $v\\in S\\setminus\n S_\\infty$ was not included in \\cite{se3} but is necessary\n\\cite{Ra1}*{p.~109 and (6.2)}.}\nthat if $\\mathbf G$ is simply connected and absolutely almost simple, then\n\\begin{equation}\n\\label{eqnSerreConjecture}\n\\text{$C(S,\\mathbf G)$ is finite if $S\\text{-rank}\\: \\mathbf G \\geq 2$ and $k_v\\text{-rank}\\: \\mathbf G>0$ for\n all $v\\in S\\setminus S_\\infty$.}\n\\end{equation}\nWhen $k$ is a number field, the main theorems in Raghunathan's papers\n\\citelist{\\cite{Ra1} \\cite{Ra2}} established the conjecture when $k\\text{-rank}\\: \\mathbf G\n> 0$ (see also \\cite{PrasadOnRaghunathan}). For a general global field,\nPrasad and Raghunathan \\cite{pr}*{Theorem~ 2.6} established the conjecture\nwhen $k\\text{-rank}\\: \\mathbf G > 0$ provided $C(S,\\mathbf G)$ is central in $\\widehat G(a)$; in\nfact they showed \\cite{pr}*{Theorems~ 2.9, 3.4} then that $C(S,\\mathbf G)$ is a\nquotient of $\\mu(k)$ provided in addition that the Kneser-Tits conjecture%\n\\footnote{Let $\\mathbf G(k)^+$ denote the subgroup of $\\mathbf G(k)$ generated by\n $k$-rational points of the unipotent radicals of the parabolic\n $k$-subgroups of $\\mathbf G$. The Kneser-Tits conjecture states that if $\\mathbf G$ is\n simply connected, almost $k$-simple, with $k\\text{-rank}\\: \\mathbf G>0$, then\n $\\mathbf G(k)^+=\\mathbf G(k)$.}\nholds for global fields. The centrality of $C(S,\\mathbf G)$ was proved when\n$k\\text{-rank}\\:\\mathbf G>0$ by Raghunathan \\citelist{\\cite{Ra1} \\cite{Ra2}} (again\nassuming that the Kneser-Tits conjecture holds) and the Kneser-Tits\nconjecture for global fields has since been demonstrated \\cite{Gille}.\nThus \\eqref{eqnSerreConjecture} holds for global fields when $k\\text{-rank}\\: \\mathbf G >\n0$; for the progress on groups with $k\\text{-rank}\\: \\mathbf G=0$ see the survey by\nRapinchuk \\cite{R2}.\n\nSerre \\cite{se3} also conjectures that $C(S,\\mathbf G)$ is infinite if $S\\text{-rank}\\: \\mathbf G\n= 1$ and verifies this for $\\mathbf G = \\SL_2$. In fact for $\\SL_2$ over ${\\mathbb Q}$,\n$C(S,\\mathbf G)$ is a free profinite group on a countable number of\ngenerators \\cite{Melnikov2}, and over a quadratic imaginary\nfield it has a finite index subgroup of this type \\cite{Lubotzky0}.\n\n\\subsection{Connection with elementary matrices}\n\\label{ssectElementaryMatrices}\n\nOur goal is a topological interpretation of the congruence subgroup kernel.\nFor this we will use the relationship of $C(S,\\mathbf G)$ with\n``elementary'' matrices. More precisely, for any $S$-arithmetic subgroup\n$\\Gamma$ let\n\\begin{equation*}\nE\\Gamma \\subset \\Gamma\n\\end{equation*}\nbe the subgroup generated by the elements of $\\Gamma$ belonging to the\nunipotent radical of any parabolic $k$-subgroup of $\\mathbf G$. As $\\Gamma$ runs\nthrough the family of\n$S$-congruence subgroups $\\Gamma({\\mathfrak a})$, we obtain a family\n$\\{E\\Gamma({\\mathfrak a})\\}_{{\\mathfrak a}\\subseteq {\\mathcal O}}$ of normal subgroups of $\\mathbf G({\\mathcal O})$ which define a\ntopology ${\\mathcal T}_e$ on $\\mathbf G({\\mathcal O})$. We denote by $\\widehat G({\\mathcal O},e)$ the\ncompletion of $\\mathbf G({\\mathcal O})$ in the topology ${\\mathcal T}_e$. For any ideal ${\\mathfrak a}\n\\subseteq {\\mathcal O}$ consider the exact sequence\n\\begin{equation*}\n1 \\ \\rightarrow\\ \\Gamma({\\mathfrak a})\/E\\Gamma({\\mathfrak a})\\ \\rightarrow\n\\ \\mathbf G({\\mathcal O})\/E\\Gamma({\\mathfrak a})\\ \\rightarrow\\ \\mathbf G({\\mathcal O})\/\\Gamma({\\mathfrak a})\n\\ \\rightarrow\\ 1\\ .\n\\end{equation*}\nTaking projective limits over the ideals ${\\mathfrak a}$ we obtain\n\\begin{equation*}\n1 \\ \\rightarrow\\ CG(e,c)\\ \\rightarrow\n\\widehat{G}({\\mathcal O},e)\\ \\rightarrow\\ \\widehat{G}({\\mathcal O},c) \\ \\rightarrow\\ 1\\ ,\n\\end{equation*}\nwhere $CG(e,c)$ is defined to be the kernel of the map on the right and\nRaghunathan's Main Lemma is used to prove that this map is surjective\n\\cite{Ra1}*{(1.21)}.\n\nAssume now that $k\\text{-rank}\\: \\mathbf G > 0$ and $S\\text{-rank}\\: \\mathbf G \\ge 2$. Then $E\\Gamma({\\mathfrak a})$ is\n$S$-arithmetic \\citelist{\\cite{Margulis} \\cite{Ra2}*{Theorem~ A, Corollary~\n 1}} (see also \\cite{Venkataramana}) and any $S$-arithmetic subgroup\n$\\Gamma$ contains $E\\Gamma({\\mathfrak a})$ for some ${\\mathfrak a}\\neq 0$ \\cite{Ra1}*{(2.1)}. So\nunder this condition, the topologies ${\\mathcal T}_e$ and ${\\mathcal T}_a$\nare the same,\n\\begin{equation*}\n\\widehat{G}({\\mathcal O},e)\\ \\cong\\ \\widehat{G}({\\mathcal O},a)\\ ,\n\\end{equation*}\nand thus\n\\begin{equation}\n\\label{eqnCongruenceKernel}\nC(S,\\mathbf G)\\ \\cong\\ CG(e,c)\\ \\cong\\ \\varprojlim_{\\mathfrak a}\n\\Gamma({\\mathfrak a})\/E\\Gamma({\\mathfrak a})\\ .\n\\end{equation}\nThis characterization of $C(S,\\mathbf G)$ will enable us to give a topological\nrealization.\n\n\\subsection{A topological realization of $C(S,\\mathbf G)$}\n\nIn this paper, our aim is to show that the algebraically and arithmetically\ndefined group $C(S,\\mathbf G)$ also has a topological interpretation as the\nfundamental group of certain compactifications of a locally symmetric\nspace. More precisely, we consider a connected, absolutely almost simple,\nsimply connected algebraic group $\\mathbf G$ defined over $k$. Let $\\mathbf H$\ndenote the restriction of scalars $\\operatorname{Res}_{k\/{\\mathbb Q}} \\mathbf G$ of $\\mathbf G$;\nthis is a group defined over ${\\mathbb Q}$ with ${\\mathbb Q}\\text{-rank}\\: \\mathbf H = k\\text{-rank}\\: \\mathbf G$.\nLet $X_\\infty =\\mathbf H({\\mathbb R})\/K$ be the symmetric space associated to\n$\\mathbf H$, where $K$ is a maximal compact subgroup of $\\mathbf H({\\mathbb R})$,\nand for $v\\in S\\setminus S_\\infty$, let $X_v$ be the Bruhat-Tits building\nof $\\mathbf G(k_v)$.\n\nConsider $X = X_\\infty \\times \\prod_{v\\in S\\setminus S_\\infty} X_v$. By\ngeneralizing the work of Borel and Serre \\citelist{\\cite{Borel-Serre}\n\\cite{BS2}} and of Zucker \\cite{Zu1}, we define in\n\\S\\S\\ref{subsectRBSarith}, \\ref{subsectRBSSarith} the reductive Borel-Serre\nbordification $\\overline{X}^{RBS}$ of $X$. For an $S$-arithmetic\nsubgroup $\\Gamma$ of $\\mathbf G(k)$, the action of $\\Gamma$ on $X$ by left translation\nextends to $\\overline{X}^{RBS}$ and the quotient\n$\\Gamma\\backslash\\overline{X}^{RBS}$ is a compact Hausdorff topological\nspace, called the \\emph{reductive Borel-Serre compactification} of\n$\\Gamma\\backslash X$. Our main result (Theorem ~\\ref{thmMainArithmetic}) is the\ncomputation of the fundamental group of\n$\\Gamma\\backslash\\overline{X}^{RBS}$. Under the mild condition that $\\Gamma$\nis a neat $S$-arithmetic group, we show (Corollary ~\\ref{corNeat}) that\n\\begin{equation}\n\\pi_1(\\Gamma\\backslash\\overline{X}^{RBS}) \\cong \\Gamma \/ E\\Gamma\n\\end{equation}\nIf $k\\text{-rank}\\: \\mathbf G >0$ and $S\\text{-rank}\\: \\mathbf G \\ge 2$ this is finite and we\nconclude from \\eqref{eqnCongruenceKernel} that \n\\begin{equation}\nC(S,\\mathbf G) \\cong \\varprojlim_{\\mathfrak a} \\pi_1(\\Gamma({\\mathfrak a})\\backslash\\overline{X}^{RBS}).\n\\end{equation}\nIn fact we show (Corollary ~\\ref{corIdentifyCSG}) that $C(S,\\mathbf G)$ is\nprecisely $\\pi_1(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS})$ for ${\\mathfrak a}$ small,\nwhere $\\Gamma^*({\\mathfrak a})$, defined by Raghunathan \\cite{Ra1}, is the smallest\n$S$-congruence subgroup containing $E\\Gamma({\\mathfrak a})$.\n\nFrom the point of view of identifying the congruence subgroup kernel $C(S,\n\\mathbf G)$, we see that $\\Gamma\\backslash \\overline{X}^{RBS}$ is the most natural\ncompactification of $\\Gamma\\backslash X$. On the other hand, the Satake\ncompactifications of the locally symmetric space $\\Gamma\\backslash X_\\infty$\nare important as well, as mentioned at the beginning of this introduction.\nIn \\S\\ref{subsectSatakeSArith} we define compactifications $\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ of $\\Gamma \\backslash X$ which generalize the\nSatake compactifications of $\\Gamma\\backslash X_\\infty$ and in\n\\S\\ref{sectFundGrpArithmetic} we calculate that their fundamental groups\nare a certain quotient of $\\pi_1(\\Gamma\\backslash\\overline{X}^{RBS})$.\n\n\\subsection{Connection to bounded generation}\nAlthough not directly addressed by this paper, we close this introduction\nby mentioning the relation of the congruence subgroup problem to the notion\nof bounded generation. A fundamental result of Borel and Harish-Chandra\n\\cite{BorelHarishChandra} is that arithmetic subgroups of algebraic groups\nare finitely generated. The proof of Borel and Harish-Chandra is in fact\nconstructive, and Grunewald and Segal \\cite{GrunewaldSegal1} have shown how\nto use it to find generators. If one assumes that the algebraic group is\nreductive then this result extends to $S$-arithmetic subgroups and in fact\n$S$-arithmetic subgroups of reductive algebraic groups are even finitely\npresented \\citelist{\\cite{BS2}*{Th\\'eor\\`eme~6.2} \\cite{GrunewaldSegal2}}.\nNote that $S$-arithmetic subgroups of a general algebraic group need not be\neven finitely generated. For example, ${\\mathbb Z}[1\/p]$ is a\n$\\{p,\\infty\\}$-arithmetic subgroup of $\\mathbf G_a$ over ${\\mathbb Q}$ and is not finitely\ngenerated.\n\nA finitely generated group $\\Gamma$ has \\emph{bounded generation} if there\nexist elements $\\gamma_1,\\gamma_2,\\dots,\\gamma_m\\in \\Gamma$ (not necessarily\ndistinct) such that any $\\gamma\\in \\Gamma$ can be written in the form\n\\begin{equation*}\n\\gamma= \\gamma_1^{k_1} \\dots \\gamma_m^{k_m}\n\\end{equation*}\nwith $k_1 , \\dots , k_m \\in {\\mathbb Z}$. The least possible value of $m$ is called\nthe \\emph{degree of bounded generation}.\n\nA free group on more than one generator does not have bounded generation.\nSince $\\SL_2({\\mathbb Z})$ contains a free group of finite index on two generators\n(for example, the commutator subgroup), it follows that it does not have\nbounded generation \\cite{Murty}*{\\S5}. Rapinchuk \\cite{R1} conjectures that\nif $\\mathbf G$ is simple and the $S$-rank of $\\mathbf G$ is $\\geq 2$, then $\\mathbf G({\\mathcal O})$\nhas bounded generation.\n\nThe relation between bounded generation and the congruence subgroup problem\nhas been clarified by recent work of Platonov and Rapinchuk\n\\cite{PlatonovRapinchuk2} and independently by Lubotzky \\cite{Lubotzky}.\nLet $T$ be the (finite) set of primes $v$ where $\\mathbf G(k_v)$ is anisotropic\nand assume that $S\\cap T=\\emptyset$. Suppose every non-central normal\nsubgroup of $\\mathbf G({\\mathcal O})$ is the inverse image of an open normal subgroup\nunder the map\n\\begin{equation*}\n\\mathbf G (k) \\to \\prod_{v\\in T} \\mathbf G(k_v) \\ .\n\\end{equation*}\nThen if $\\mathbf G({\\mathcal O})$ has bounded generation they prove that $C(S,\\mathbf G)$ is\nfinite.\n\nThus another way to establish that $C(S,\\mathbf G)$ is finite is to show that\n$\\mathbf G({\\mathcal O})$ has bounded generation. For example, Tavgen\\cprime\\\n\\cite{tavgen} has established that $\\mathbf G({\\mathcal O})$ has bounded generation for\n$k$-simple groups $\\mathbf G$ which are quasi-split over $k$ with $k$-rank $\\ge 2$\n(except possibly for type ${}^6D_4$). In another direction, if $|S|$ is\nassumed sufficiently large (depending only on $[ k:{\\mathbb Q} ]$), Murty and\nLoukanidis have proved bounded generation for $\\SL_n({\\mathcal O})$, $n\\ge 2$, and\n$\\SP_{2n}({\\mathcal O})$, $n\\ge 1$; this work is announced in \\cite{Murty} and\npartially included in the thesis of Loukanidis \\cite{Loukanidis}. The\nproof, which uses analytic number theory, actually gives an explicit bound\non the degree of bounded generation depending only on $[k:{\\mathbb Q}]$; bounds on\nthe degree which depend also on the discriminant of $k$ have been obtained\npreviously by other authors.\n\n\\subsection{Other directions}\n\n\\subsubsection{Infinite $C(S,\\mathbf G)$}\n\nThis paper has focused on the case $S\\text{-rank}\\: \\mathbf G \\ge 2$ where Serre's\nconjecture says that $C(S,\\mathbf G)$ is finite. It would be interesting to\ninvestigate topological interpretations in the case $S\\text{-rank}\\: \\mathbf G = 1$ and\n$C(S,\\mathbf G)$ is infinite.\n\n\\subsubsection{Function fields}\n\nUsually the congruence subgroup problem is considered for algebraic groups\ndefined over global fields, not just algebraic number fields as considered\nhere. As noted in \\S\\ref{ssectKnownResults}, for $k$ a global field, the\ncongruence subgroup kernel $C(S, \\mathbf G)$ is finite for $\\mathbf G$ simply connected,\nabsolutely almost simple with $k\\text{-rank}\\: \\mathbf G >0$ and $S\\text{-rank}\\: \\mathbf G\\ge 2$. A\nnatural question is to give a topological interpretation in this case as\nwell. Here there are no infinite places so it seems plausible to consider\nthe fundamental group of suitable compactifications of an $S$-arithmetic\nquotient of the product of Bruhat-Tits buildings $\\prod_{v\\in S} X_{v}$.\nSeveral compactifications of Bruhat-Tits buildings have been considered: the\nBorel-Serre compactification in which the spherical Tits building is placed\nat infinity \\cite{BS2}; a polyhedral compactification due to Landvogt\n\\cite{Landvogt}; and compactifications\nassociated to linear representations \\citelist{\\cite{Werner}\n\\cite{RemyThuillierWernerI} \\cite{RemyThuillierWernerII}}. These last \ncompactifications are analogous to the Satake\ncompactifications of symmetric spaces and recover Landvogt's\ncompactification as a special case for the generic representation; thus\nLandvogt's compactification is analogous to the maximal Satake\ncompactification. It would be interesting to see if there is an analogy of\nSatake's theory of rational boundary components which would lead to\ncorresponding compactifications of the $S$-arithmetic quotients.\n\n\\section{The reductive Borel-Serre and Satake compactifications: the\n arithmetic case}\n\\label{sectCompactificationsArithmetic}\n\nIn order to establish notation and set the framework for later proofs, we\nrecall in \\S\\S\\ref{ssectBSarith}--\\ref{subsectSatakeArith} several natural\ncompactifications of the locally symmetric space $\\Gamma\\backslash X_\\infty$\nassociated to an arithmetic group $\\Gamma$; in each case a bordification of\n$X_\\infty$ is described on which $\\mathbf G(k)$ acts. We also examine the\nstabilizer subgroups of points in these bordifications. The case of\ngeneral $S$-arithmetic groups will be treated in\n\\S\\ref{sectCompactificationsSArithmetic}. Throughout the paper, $\\mathbf G$ will\ndenote a connected, absolutely almost simple, simply connected algebraic\ngroup defined over a number field $k$.\n\n\\subsection{Proper and discontinuous actions}\n\\label{ssectProperDiscontinuousActions}\nRecall \\cite{BourbakiTopologiePartOne}*{III, \\S4.4, Prop.~7} that a\ndiscrete group $\\Gamma$ acts \\emph{properly} on a Hausdorff space $Y$ if and\nonly if for all $y$, $y'\\in Y$, there exist neighborhoods $V$ of $y$ and\n$V'$ of $y'$ such that $\\gamma V\\cap V'\\neq \\emptyset$ for only finitely\nmany $\\gamma \\in \\Gamma$. We will also need the following weaker condition on\nthe group action:\n\n\\begin{defi}[\\cite{Gro}*{Definition~1}]\n\\label{defnDiscontinuous}\nThe action of a discrete group $\\Gamma$ on a topological space $Y$ is\n\\emph{discontinuous} if\n\\begin{enumerate}\n\\item\\label{itemDiscontinuousTwoPoints} for all $y$, $y'\\in Y$ with\n $y'\\notin \\Gamma y$ there exists neighborhoods $V$ of $y$ and $V'$ of $y'$\n such that $\\gamma V\\cap V' =\\emptyset$ for all $\\gamma\\in \\Gamma$, and\n\\item\\label{itemDiscontinuousOnePoint} for all $y\\in Y$ there exists a\n neighborhood $V$ of $y$ such that $\\gamma V\\cap V = \\emptyset$ for\n $\\gamma \\notin \\Gamma_y$ and $\\gamma V = V$ for $\\gamma \\in \\Gamma_y$.\n\\end{enumerate}\n\\end{defi}\n\nIt is easy to check that a group action is proper if and only if it is\ndiscontinuous and the stabilizer subgroup $\\Gamma_y$ is finite for all $y\\in\nY$.\n\n\\subsection{The locally symmetric space associated to an arithmetic subgroup}\nLet $S_{\\infty}$ be the set of all\ninfinite places of $k$. For each $v\\in S_\\infty$, let $k_{v}$ be the\ncorresponding completion of $k$ with respect to a norm associated with $v$;\nthus either $k_{v}\\cong {\\mathbb R}$ or $k_{v}\\cong {\\mathbb C}$. For each $v\\in\nS_{\\infty}$, $\\mathbf G(k_{v})$ is a (real) Lie group.\n\nDefine $G_{\\infty}=\\prod_{v\\in S_{\\infty}}\\mathbf G(k_{v})$, a semisimple Lie\ngroup with finitely many connected components. Fix a maximal compact\nsubgroup $K$ of $G_{\\infty}$. When endowed with a $G$-invariant metric,\n$X_\\infty = G_{\\infty}\/K$ is a Riemannian symmetric space of noncompact\ntype and is thus contractible. Embed $\\mathbf G(k)$ into $G_\\infty$ diagonally.\nThen any arithmetic subgroup $\\Gamma\\subset \\mathbf G(k)$ is a discrete subgroup of\n$G_\\infty$ and acts properly on $X_\\infty$. It is known that the quotient\n$\\Gamma\\backslash X_\\infty$ is compact if and only if the $k$-rank of $\\mathbf G$ is\nequal to 0. In the following, we assume that the $k$-rank of $\\mathbf G$ is\npositive so that $\\Gamma\\backslash X_\\infty$ is noncompact.\n\nSince the theory of compactifications of locally symmetric spaces is\nusually expressed in terms of algebraic groups defined over ${\\mathbb Q}$, let\n$\\mathbf H=\\operatorname{Res}_{k\/{\\mathbb Q}}\\mathbf G$ be the algebraic group defined over\n${\\mathbb Q}$ obtained by restriction of scalars; it satisfies\n\\begin{equation}\n\\label{eqnPointsOfRestrictionScalars}\n\\mathbf H({\\mathbb Q})=\\mathbf G(k) \\quad\\text{and}\\quad \\mathbf H(\\mathbb R)=G_{\\infty}\\ .\n\\end{equation}\nThe space $X_{\\infty}$ can be identified with the symmetric space of\nmaximal compact subgroups of $\\mathbf H(\\mathbb R)$, $X_{\\infty}=\\mathbf\nH(\\mathbb R)\/K$, and the arithmetic subgroup $\\Gamma\\subset \\mathbf G(k)$ corresponds\nto an arithmetic subgroup $\\Gamma\\subset \\mathbf H({\\mathbb Q})$. Restriction of\nscalars yields a one-to-one correspondence between parabolic $k$-subgroups\nof $\\mathbf G$ and parabolic ${\\mathbb Q}$-subgroups of $\\mathbf H$ so that the analogue of\n\\eqref{eqnPointsOfRestrictionScalars} is satisfied.\n\n\\subsection{The Borel-Serre compactification}\n\\label{ssectBSarith}\n(For details see the original paper \\cite{Borel-Serre}, as well as\n\\cite{Borel-Ji}.) For each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$,\nconsider the Levi quotient $\\mathbf L_{\\P} = \\P\/\\mathbf N_{\\P}$ where $\\mathbf N_{\\P}$ is\nthe unipotent radical of $\\P$. This is a reductive group defined over\n${\\mathbb Q}$. There is an almost direct product $\\mathbf L_{\\P} = \\mathbf S_{\\P}\n\\cdot \\mathbf M_{\\P}$, where $\\mathbf S_{\\P}$ is the maximal ${\\mathbb Q}$-split\ntorus in the center of $\\mathbf L_{\\P}$ and $\\mathbf M_{\\P}$ is the\nintersection of the kernels of the squares of all characters of $\\mathbf\nL_{\\P}$ defined over ${\\mathbb Q}$. The real locus $L_P= \\mathbf L_\\P({\\mathbb R})$ has a\ndirect product decomposition $A_P \\cdot M_P$, where $A_P = \\mathbf\nS_\\P({\\mathbb R})^0$ and $M_P = \\mathbf M_\\P({\\mathbb R})$. The dimension of $A_P$ is called\nthe \\emph{parabolic ${\\mathbb Q}$-rank} of $\\P$.\n\nThe real locus $P=\\P({\\mathbb R})$ has a Langlands decomposition\n\\begin{equation}\\label{rationalLanglands}\nP=N_{P} \\ltimes (\\widetilde A_P \\cdot \\widetilde M_ P),\n\\end{equation}\nwhere $N_{P}= \\mathbf N_{\\P}({\\mathbb R})$ and $\\widetilde A_P \\cdot \\widetilde M_ P$ is\nthe lift of $A_P \\cdot M_P$ to the unique Levi subgroup of $P$ which is\nstable under the Cartan involution $\\theta$ associated with $K$.\n\nSince $P$ acts transitively on $X_\\infty$, the Langlands decomposition induces a\nhorospherical decomposition\n\\begin{equation}\\label{horo}\nX_\\infty \\cong A_P\\times N_{P}\\times X_P,\\quad u\\tilde a\\tilde mK \\mapsto\n(\\tilde a,u,\\tilde m(K\\cap \\widetilde M_P),\n\\end{equation}\nwhere \n\\begin{equation*}\nX_P= \\widetilde M_P \/ (K \\cap \\widetilde M_P) \\cong L_P\/(A_P\\cdot K_P)\n\\end{equation*}\nis a symmetric space (which might contain an Euclidean factor) and is\ncalled the \\emph{boundary symmetric space associated with $\\P$}. The\nsecond expression for $X_P$ is preferred since $\\mathbf L_\\P$ is defined\nover ${\\mathbb Q}$; here $K_P\\subseteq \\mathbf M_\\P({\\mathbb R})$ corresponds to $K \\cap\n\\widetilde M_P$\n\nFor each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, define the\nBorel-Serre boundary component\n\\begin{equation*}\ne(P)=N_{P}\\times X_P\n\\end{equation*}\nwhich we view as the quotient of $X_\\infty$ obtained by collapsing the\nfirst factor in \\eqref{horo}. The action of $P$ on $X_\\infty$ descends to\nan action on $e(P)=N_{P}\\times X_P$ given by\n\\begin{equation}\n\\label{eqnBoundaryAction}\np\\cdot (u, y) = (pu\\tilde m_p^{-1}\\tilde a_p^{-1} , \\tilde a_p \\tilde m_p\ny), \\qquad \\text{for $p=u_p \\tilde a_p \\tilde m_p\\in P$.}\n\\end{equation}\nDefine the Borel-Serre partial compactification $\\overline{X}_\\infty^{BS}$ (as a\n\\emph{set}) by\n\\begin{equation}\n\\label{BSPartialCompactification}\n\\overline{X}_\\infty^{BS}=X_\\infty\\cup \\coprod_{\\P\\subset \\mathbf H} e(P).\n\\end{equation}\n\nLet $\\Delta_P$ be the simple ``roots'' of the adjoint action of $A_P$ on\nthe Lie algebra of $N_P$ and identify $A_P$ with $({\\mathbb R}^{>0})^{\\Delta_P}$ by\n$a \\mapsto (a^{-\\alpha})_{\\alpha\\in\\Delta_P}$. Enlarge $A_P$ to the\ntopological semigroup $\\overline A_P \\cong ({\\mathbb R}^{\\ge0})^{\\Delta_P}$ by\nallowing $a^\\alpha$ to attain infinity and define\n\\begin{equation*}\n\\overline A_P(s) = \\{\\, a\\in \\overline A_P\\mid a^{-\\alpha} < s^{-1} \\text{\n for all $\\alpha\\in \\Delta_P$}\\,\\}\\cong [0,s^{-1})^{\\Delta_P}\\ ,\\qquad\n \\text{for $s>0$}\\ .\n\\end{equation*}\nSimilarly enlarge the Lie algebra ${\\mathfrak a}_P \\subset \\overline{\\mathfrak a}_P$. The\ninverse isomorphisms $\\exp\\colon {\\mathfrak a}_P \\to A_P$ and $\\log\\colon A_P \\to\n{\\mathfrak a}_P$ extend to isomorphisms\n\\begin{equation*}\n\\overline A_P \\overset{\\log}{\\longrightarrow} \\overline {\\mathfrak a}_P\n\\qquad\\text{and} \\qquad \\overline {\\mathfrak a}_P \\overset{\\exp}{\\longrightarrow}\n\\overline A_P.\n\\end{equation*}\n\nTo every parabolic ${\\mathbb Q}$-subgroup $\\mathbf Q\\supseteq \\P$ there corresponds\na subset $\\Delta_P^Q \\subseteq \\Delta_P$ and we let $o_Q\\in\n\\overline A_P$ be the point with coordinates $o_Q^{-\\alpha} =1$\nfor $\\alpha\\in \\Delta_P^Q$ and $o_Q^{-\\alpha} =0$ for\n$\\alpha\\notin \\Delta_P^Q$. Then $\\overline A_P = \\coprod_{\\mathbf Q\n\\supseteq \\P} A_P \\cdot o_Q$ is the decomposition into\n$A_P$-orbits.\n\nDefine the \\emph{corner associated to $\\mathbf P$} to be\n\\begin{equation}\n\\label{Pcorner}\nX_\\infty(P) = \\overline A_P \\times e(P) = \\overline A_P \\times N_P \\times X_P.\n\\end{equation}\nWe identify $e(Q)$ with the subset $ (A_P\\cdot o_Q) \\times N_P\\times X_P$.\nIn particular, $e(P)$ is identified with the subset $\\{o_P\\}\\times\nN_P\\times X_P$ and $X_\\infty$ is identified with the open subset $A_P \\times\nN_P\\times X_P \\subset X_\\infty(P)$ (compare \\eqref{horo}). Thus we have a\nbijection\n\\begin{equation}\n\\label{strataPcorner}\nX_\\infty(P) \\cong X_\\infty \\cup \\coprod_{\\P \\subseteq \\mathbf Q \\subset\n \\mathbf H} e(Q).\n\\end{equation}\n\nNow give $\\overline X_\\infty^{BS}$ the finest topology so that for all\nparabolic ${\\mathbb Q}$-subgroups $\\P$ of $\\mathbf H$ the inclusion of\n\\eqref{strataPcorner} into \\eqref{BSPartialCompactification} is a\ncontinuous inclusion of an open subset. Under this topology, a sequence\n$x_n\\in X$ converges in $\\overline X_\\infty^{BS}$ if and only if there\nexists a parabolic ${\\mathbb Q}$-subgroup $\\P$ such that if we write $x_n=(a_n, u_n,\ny_n)$ according to the decomposition of \\eqref{horo}, then $(u_n,y_n)$\nconverges to a point in $e(P)$ and $a_n^\\alpha\\to \\infty$ for all\n$\\alpha\\in \\Delta_P$. The space $\\overline X_\\infty^{BS}$ is a manifold\nwith corners. It has the same homotopy type as $X_\\infty$ and is thus\ncontractible \\cite{Borel-Serre}.\n\nThe action of $\\mathbf H({\\mathbb Q})$ on $X_\\infty$ extends to a continuous action\non $\\overline{X}_\\infty^{BS}$ which permutes the boundary components:\n$g\\cdot e(P) = e(gPg^{-1})$ for $g\\in \\mathbf H({\\mathbb Q})$. The normalizer of\n$e(P)$ is $\\P({\\mathbb Q})$ which acts according to \\eqref{eqnBoundaryAction}.\n\nIt is shown in \\cite{Borel-Serre} that the action of $\\Gamma$ on\n$\\overline{X}_\\infty^{BS}$ is proper and the quotient $\\Gamma\\backslash\n\\overline{X}_\\infty^{BS}$, the \\emph{Borel-Serre compactification}, is a compact\nHausdorff space. It is a manifold with corners if $\\Gamma$ is torsion-free.\n\n\\subsection{The reductive Borel-Serre compactification}\n\\label{subsectRBSarith}\nThis compactification was first constructed by Zucker \\cite{Zu1}*{\\S4} (see also\n\\cite{GHM}). For each parabolic ${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, define\nits reductive Borel-Serre boundary component $\\hat{e}(P)$ by\n\\begin{equation*}\n\\hat{e}(P)=X_P\n\\end{equation*}\nand set\n\\begin{equation*}\n\\overline{X}_\\infty^{RBS}=X_\\infty\\cup \\coprod_{\\P} \\hat{e}(P).\n\\end{equation*}\nThe projections $p_P\\colon e(P) = N_P\\times X_P \\to \\hat e(P) = X_P$ induce\na surjection $p\\colon \\overline{X}_\\infty^{BS} \\to\n\\overline{X}_\\infty^{RBS}$ and we give $\\overline{X}_\\infty^{RBS}$ the\nquotient topology. Its topology can also be described in terms of\nconvergence of interior points to the boundary points via the horospherical\ndecomposition in equation \\eqref{horo}. Note that $\\overline{X}^{RBS}$ is\nnot locally compact, although it is compactly generated (being a Hausdorff\nquotient of the locally compact space $\\overline{X}^{BS}$). The action of\n$\\mathbf H({\\mathbb Q})$ on $\\overline{X}_\\infty^{BS}$ descends to a continuous\naction on $\\overline{X}_\\infty^{RBS}$.\n\n\\begin{lem}\n\\label{lemStabilizersRBS}\nLet $\\P$ be a parabolic ${\\mathbb Q}$-subgroup of $\\mathbf H$.\nThe stabilizer $\\mathbf H({\\mathbb Q})_z= \\mathbf G(k)_z$ of $z\\in X_P$ under the action of\n$\\mathbf H({\\mathbb Q})$ on $\\overline{X}^{RBS}_\\infty$ satisfies a short exact sequence\n\\begin{equation*}\n1 \\to \\mathbf N_{\\P}({\\mathbb Q}) \\to \\mathbf H({\\mathbb Q})_z \\to \\mathbf L_{\\P}({\\mathbb Q})_z \\to 1\n\\end{equation*}\nwhere $\\mathbf L_{\\P}({\\mathbb Q})_z$ is the stabilizer of $z$ under the action of\n$\\mathbf L_{\\P}({\\mathbb Q})$ on $X_P$.\n\\end{lem}\n\\begin{proof}\nThe normalizer of $X_P$ under the action of $\\mathbf H({\\mathbb Q})$ is $\\P({\\mathbb Q})$\nwhich acts via its quotient $\\mathbf L_{\\P}({\\mathbb Q})$. \n\\end{proof}\n\nBy the lemma, the action of $\\Gamma$ on $\\overline{X}_\\infty^{RBS}$ is not\nproper since the stabilizer of a boundary point in $X_P$ contains the\ninfinite group $\\Gamma_{N_P} = \\Gamma\\cap N_P$. Nonetheless\n\\begin{lem}\n\\label{lemRBSDiscontinuous}\nThe action of an arithmetic subgroup $\\Gamma$ on $\\overline{X}_\\infty^{RBS}$\nis discontinuous and the arithmetic quotient\n$\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ is a compact Hausdorff space.\n\\end{lem}\n\n\\begin{proof}\nWe begin by verifying Definition\n~\\ref{defnDiscontinuous}\\ref{itemDiscontinuousOnePoint}. Let $x\\in X_P\n\\subseteq \\overline{X}_\\infty^{RBS}$. Set $\\Gamma_P = \\Gamma\\cap P$ and\n$\\Gamma_{L_P} = \\Gamma_P\/\\Gamma_{N_P}$. Since $\\Gamma_{L_P}$ acts properly on $X_P$\nthere exists a neighborhood $O_x$ of $x$ in $X_P$ such that $\\bar \\gamma\nO_x \\cap O_x \\neq \\emptyset$ if and only if $\\bar \\gamma \\in \\Gamma_{L_P,x}$,\nin which case $\\bar \\gamma O_x = O_x$. We can assume $O_x$ is relatively\ncompact. Set $V=p(\\overline{A}_P(s)\\times N_P \\times O_x)$, where we chose\n$s$ sufficiently large so that that only identifications induced by $\\Gamma$\non $V$ already arise from $\\Gamma_P$ \\cite{Zu3}*{(1.5)}. Thus $\\gamma V\\cap\nV\\neq \\emptyset$ if and only if $\\gamma \\in \\Gamma_P$ and $\\gamma \\Gamma_{N_P}\n\\in \\Gamma_{L_P,x}$; by Lemma ~\\ref{lemStabilizersRBS} this occurs if and only\nif $\\gamma \\in \\Gamma_x$ as desired.\n\nTo verify Definition\n~\\ref{defnDiscontinuous}\\ref{itemDiscontinuousTwoPoints} we will show the\nequivalent condition that $\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ is\nHausdorff (compare \\cite{Zu1}*{(4.2)}). Compactness will follow since it\nis the image of a compact space under the induced projection $p'\\colon\n\\Gamma\\backslash \\overline{X}_\\infty^{BS} \\to \\Gamma\\backslash\n\\overline{X}_\\infty^{RBS}$. Observe that $p'$ is a quotient map and that\nits fibers, each being homeomorphic to $\\Gamma_{N_P}\\backslash N_P$ for some\n$\\P$, are compact. For $y\\in \\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ and\n$W$ a neighborhood of $p'^{-1}(y)$, we claim there exists $U\\ni y$ open\nsuch that $p'^{-1}(U)\\subseteq W$. This suffices to establish Hausdorff,\nfor if $y_1\\neq y_2 \\in \\Gamma\\backslash\\overline{X}_\\infty^{RBS}$ and $W_1$\nand $W_2$ are disjoint neighborhoods of the compact fibers $p'^{-1}(y_1)$\nand $p'^{-1}(y_2)$, there must exist $U_1$ and $U_2$, neighborhoods of\n$y_1$ and $y_2$, such that $p'^{-1}(U_i) \\subseteq W_i$ and hence $U_1\\cap\nU_2 =\\emptyset$.\n\nTo prove the claim, choose $x\\in X_P$ such that $y=\\Gamma x$. Let $q\\colon\n\\overline{X}_\\infty^{BS} \\to \\Gamma\\backslash \\overline{X}_\\infty^{BS} $ be\nthe quotient map. The compact fiber $p'^{-1}(y)$ may be covered by\nfinitely many open subsets $q(\\overline A_P(s_\\mu)\\times C_{P,\\mu} \\times\nO_{P,\\mu}) \\subseteq W$ where $C_{P,\\mu} \\subseteq N_P$ and $x\\in\nO_{P,\\mu}\\subseteq X_P$. Define a neighborhood $V$ of the fiber by\n\\begin{equation*}\np'^{-1}(y) \\subset V = q(\\overline A_P(s)\\times C_{P}\n\\times O_{P}) \\subseteq W\n\\end{equation*}\nwhere $s = \\max \\,s_\\mu$, $O_P = \\bigcap O_{P,\\mu}$, and $C_P = \\bigcup C_{P,\\mu}$.\nSince $\\Gamma_{N_P}C_P = N_P$, we see $V=p'^{-1}(U)$ for some $U\\ni y$ as\ndesired.\n\\end{proof}\n\n\\subsection{Satake compactifications}\n\\label{subsectSatakeArith}\nFor arithmetic quotients of $X_\\infty$, the Satake compactifications\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ form an important family of\ncompactifications. When $X_\\infty$ is Hermitian, one example is the Baily-Borel\nSatake compactification. The construction has three steps.\n\\begin{enumerate}\n\\item Begin%\n\\footnote{Here we follow \\cite{Cass} in beginning with a spherical\n representation. Satake's original construction \\cite{sat1} started with a\n non-spherical representation but then constructed a spherical\n representation by letting $G_\\infty$ act on the space of self-adjoint\n endomorphisms of $V$ with respect to an admissible inner product. See\n \\cite{sap2} for the relation of the two constructions.}\nwith a representation $(\\tau,V)$ of $\\mathbf H$ which has a nonzero\n$K$-fixed vector $v\\in V$ (a \\emph{spherical representation}) and which is\nirreducible and nontrivial on each noncompact ${\\mathbb R}$-simple factor of\n$\\mathbf H$. Define the Satake compactification $\\overline{X}_\\infty^{\\tau}$\nof $X$ to be the closure of the image of the embedding $X_\\infty \\hookrightarrow\n\\mathbb P(V)$, $gK \\mapsto [ \\tau(g) v]$. The action of $G_\\infty$ extends\nto a continuous action on $\\overline{X}_\\infty^{\\tau}$ and the set of points\nfixed by $N_P$, where $\\P$ is any parabolic ${\\mathbb R}$-subgroup, is called a\n\\emph{real boundary component}. The compactification\n$\\overline{X}_\\infty^{\\tau}$ is the disjoint union of its real boundary\ncomponents.\n\n\\item Define a partial compactification\n ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}\\subseteq \\overline{X}_\\infty^{\\tau}$\n by taking the union of $X_\\infty$ and those real boundary components that\n meet the closure of a Siegel set. Under the condition that\n $\\overline{X}_\\infty^{\\tau}$ is \\emph{geometrically rational}\n \\cite{Cass}, this is equivalent to considering those real boundary\n components whose normalizers are parabolic ${\\mathbb Q}$-subgroups; call these the\n \\emph{rational boundary components}. Instead of the subspace topology\n induced from $\\overline{X}_\\infty^{\\tau}$, give\n ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ the Satake topology \\cite{sat2}.\n\n\\item Still under the condition that $\\overline{X}_\\infty^{\\tau}$ is\n geometrically rational, one may show that the arithmetic subgroup $\\Gamma$\n acts continuously on ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ with a\n compact Hausdorff quotient, $\\Gamma\\backslash\n {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$. This is the \\emph{Satake\n compactification} of $\\Gamma\\backslash X_\\infty$.\n\\end{enumerate}\n\nThe geometric rationality condition above always holds if the\nrepresentation $(\\tau,V)$ is rational over ${\\mathbb Q}$ \\cite{sap2}. It also holds\nfor the Baily-Borel Satake compactification \\cite{BB}, as well as most\nequal-rank Satake compactifications including all those where ${\\mathbb Q}\\text{-rank}\\:\n\\mathbf H >2$.\n\nWe will now describe an alternate construction of\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ due to Zucker \\cite{Zu2}. Instead of\nthe Satake topology, Zucker gives ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ the\nquotient topology under a certain surjection $\\overline{X}_\\infty^{RBS}\n\\to {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ described below. It is this\ntopology we will use in this paper. Zucker proves that the resulting two\ntopologies on $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ coincide.\n\nLet $(\\tau,V)$ be a spherical representation as above. We assume that\n$\\overline{X}_\\infty^{\\tau}$ is geometrically rational. For any parabolic\n${\\mathbb Q}$-subgroup $\\P$ of $\\mathbf H$, let $X_{P,\\tau}\\subseteq \\overline\nX_\\infty^{\\tau}$ be the real boundary component fixed pointwise by $N_P$;\ngeometric rationality implies that $X_{P,\\tau}$ is actually a rational\nboundary component. The transitive action of $P$ on $X_{P,\\tau}$ descends\nto an action of $L_P = P\/N_P$. The geometric rationality condition ensures\nthat there exists a normal ${\\mathbb Q}$-subgroup $\\mathbf L_{\\P, \\tau} \\subseteq\n\\mathbf L_{\\P}$ with the property that $L_{P,\\tau}= \\mathbf L_{\\P,\n \\tau}({\\mathbb R})$ is contained in the centralizer\n$\\operatorname{Cent}(X_{P,\\tau})$ of $X_{P,\\tau}$ and\n$\\operatorname{Cent}(X_{P,\\tau})\/L_{P,\\tau}$ is compact. Then $X_{P,\\tau}$\nis the symmetric space associated to the ${\\mathbb Q}$-group $\\mathbf H_{\\P,\n\\tau} = \\mathbf L_{\\P} \/ \\mathbf L_{\\P,\\tau}$. There is an\nalmost direct product decomposition\n\\begin{equation}\n\\label{eqnSatakeLeviDecomposition}\n\\mathbf L_{\\P} = \\widetilde {\\mathbf H}_{\\P, \\tau} \\cdot \\mathbf L_{\\P,\n \\tau}\\ ,\n\\end{equation}\nwhere $\\widetilde {\\mathbf H}_{\\P, \\tau}$ is a lift of $\\mathbf H_{\\P,\n \\tau}$; the root systems of these factors may be described using the\nhighest weight of $\\tau$. We obtain a decomposition of symmetric spaces\n\\begin{equation}\n\\label{eqnBoundaryDecomposition}\nX_P = X_{P,\\tau} \\times W_{P,\\tau}\\ .\n\\end{equation}\n\nDifferent parabolic ${\\mathbb Q}$-subgroups can yield the same rational boundary\ncomponent $X_{P,\\tau}$; if $\\P^\\dag$ is the maximal such parabolic\n${\\mathbb Q}$-subgroup, then $P^\\dag=\\P^\\dag({\\mathbb R})$ is the normalizer of $X_{P,\\tau}$.\nThe parabolic ${\\mathbb Q}$-subgroups that arise as the normalizers of rational\nboundary components are called \\emph{$\\tau$-saturated}. For example, all\nparabolic ${\\mathbb Q}$-subgroups are saturated for the maximal Satake\ncompactification, while only the maximal parabolic ${\\mathbb Q}$-subgroups are\nsaturated for the Baily-Borel Satake compactification when $\\mathbf H$ is\n${\\mathbb Q}$-simple. In general, the class of $\\tau$-saturated parabolic\n${\\mathbb Q}$-subgroups can be described in terms of the highest weight of $\\tau$.\n\nDefine \n\\begin{equation*}\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}=X_\\infty\\cup \\coprod_{\\text{$\\mathbf Q$\n $\\tau$-saturated}} X_{Q,\\tau}\\ .\n\\end{equation*}\nA surjection $p\\colon \\overline{X}_\\infty^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is obtained by mapping $X_P$ to\n$X_{P,\\tau} = X_{P^\\dag,\\tau}$ via the projection on the first factor in\n\\eqref{eqnBoundaryDecomposition}. Give ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$\nthe resulting quotient topology; the action of $\\mathbf H({\\mathbb Q})$ on\n$\\overline{X}_\\infty^{RBS}$ descends to a continuous action on\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$. \n\nLet $\\P_\\tau$ be the inverse image of $\\mathbf L_{\\P,\\tau}$ under\nthe projection $\\P \\to \\P\/\\mathbf N_{\\P}$.\n\\begin{lem}\n\\label{lemStabilizersSatake}\nLet $\\P$ be a $\\tau$-saturated parabolic ${\\mathbb Q}$-subgroup of $\\mathbf H$. The\nstabilizer $\\mathbf H({\\mathbb Q})_z = \\mathbf G(k)_z$ of $z\\in X_{P,\\tau}$ under the\naction of $\\mathbf H({\\mathbb Q})$ on \n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ satisfies a short exact sequence\n\\begin{equation*}\n1 \\to \\P_{\\tau}({\\mathbb Q}) \\to \\mathbf H({\\mathbb Q})_z \\to \\mathbf H_{\\P,\\tau}({\\mathbb Q})_z \\to 1,\n\\end{equation*}\nwhere $\\mathbf H_{\\P,\\tau}({\\mathbb Q})_z$ is the stabilizer of $z$ under the action\nof $\\mathbf H_{\\P,\\tau}({\\mathbb Q})$ on $X_{P,\\tau}$.\n\\end{lem}\n\\begin{proof}\nAs in the proof of Lemma ~\\ref{lemStabilizersRBS}, the normalizer of\n$X_{P,\\tau}$ is $\\P({\\mathbb Q})$ which acts via its quotient $\\P({\\mathbb Q})\/\\P_\\tau({\\mathbb Q}) =\n\\mathbf H_{\\P,\\tau}({\\mathbb Q})$.\n\\end{proof}\n\nSimilarly to $\\overline{X}^{RBS}$, the space\n${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is not locally compact and $\\Gamma$ does\nnot act properly. Nonetheless one has the\n\\begin{lem}\n\\label{lemSatakeDiscontinuous}\nThe action of an arithmetic subgroup $\\Gamma$ on ${}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$\nis discontinuous and the arithmetic quotient\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ is a compact Hausdorff space.\n\\end{lem}\n\nThe proof is similar to Lemma ~\\ref{lemRBSDiscontinuous} since the fibers\nof $p'$ are again compact, being reductive Borel-Serre compactifications of\nthe $W_{P^\\dag,\\tau}$. The \\emph{Satake compactification} of\n$\\Gamma\\backslash X_\\infty$ associated to $\\tau$ is $\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$.\n\nIn the case when the representation $\\tau$ is generic one obtains the\nmaximal Satake compactification $\\overline{X}_\\infty^{\\max}$. This is\nalways geometrically rational and the associated\n${}_{\\mathbb Q}\\overline{X}_\\infty^{\\max}$ is very similar to\n$\\overline{X}_\\infty^{RBS}$. Indeed in this case $X_P = X_{P,\\tau} \\times\n({}_{\\mathbb R} A_{P}\/A_{P})$, where ${}_{\\mathbb R} A_{P}$ is defined like $A_P$ but using a\nmaximal ${\\mathbb R}$-split torus instead of a maximal ${\\mathbb Q}$-split torus, and the\nquotient map simply collapses the Euclidean factor ${}_{\\mathbb R} A_{P}\/A_{P}$ to a\npoint. In particular, if ${\\mathbb Q}\\text{-rank }\\mathbf H = {\\mathbb R}\\text{-rank }\n\\mathbf H$, then $\\Gamma\\backslash {}_{\\mathbb Q}\\overline{X}_\\infty^{\\max} \\cong\n\\Gamma\\backslash\\overline{X}_\\infty^{RBS}$.\n\n\\section{The Bruhat-Tits buildings}\n\\label{sectBruhatTitsBuildings}\n\nFor a finite place $v$, let $k_{v}$ be the completion of $k$ with respect\nto a norm associated with $v$. Bruhat and Tits \\citelist{\\cite{BruhatTits1}\n \\cite{BruhatTits2}} constructed a building $X_{v}$ which reflects the\nstructure of $\\mathbf G(k_{v})$. The building $X_{v}$ is made up of subcomplexes\ncalled \\emph{apartments} corresponding to the maximal $k_{v}$-split tori in\n$\\mathbf G$ and which are glued together by the action of $\\mathbf G(k_{v})$. We give an\noutline of the construction here together with the properties of $X_{v}$\nwhich are needed in the sections below; in addition to the original papers,\nwe benefited greatly from\n\\citelist{\\cite{ji}*{\\S3.2}\\cite{Landvogt}\\cite{Tits}}.\n\nIn this section we fix a finite place $v$ and a corresponding discrete\nvaluation $\\omega$.\n\n\\subsection{The apartment}\n\nLet $\\Split$ be a maximal $k_{v}$-split torus in $\\mathbf G$ and let\n$X^{*}(\\Split)= \\Hom_{k_{v}}(\\Split, \\mathbf G_{m})$ and $X_{*}(\\Split)\n=\\Hom_{k_{v}}(\\mathbf G_{m}, \\Split)$ denote the $k_v$-rational characters and\ncocharacters of $\\Split$ respectively. Denote by $\\Phi \\subset\nX^{*}(\\Split)$ the set of $k_{v}$-roots of $\\mathbf G$ with respect to\n$\\Split$. Let $\\N$ and $\\Cent$ denote the normalizer and the centralizer,\nrespectively, of $\\Split$; set $N=\\N(k_{v})$, $Z=\\Cent(k_{v})$. The Weyl\ngroup $W =\nN\/Z$ of $\\Phi$ acts on the real vector space\n\\begin{equation*}\nV = X_{*}(\\Split) \\otimes_{{\\mathbb Z}}{\\mathbb R} = \\Hom_{{\\mathbb Z}}(X^{*}(\\Split) , {\\mathbb R})\n\\end{equation*}\nby linear transformations; for $\\alpha\\in\\Phi$, let $r_\\alpha$ denote the\ncorresponding reflection of $V$.\n\nLet $A$ be the affine space underlying $V$ and let $\\Aff(A)$ denote the\ngroup of invertible affine transformations. We identify $V$ with the\ntranslation subgroup of $\\Aff(A)$. There is an action of $Z$ on $A$ via\ntranslations, $\\nu\\colon Z\\rightarrow V \\subset \\Aff(A)$, determined by\n\\begin{equation*}\n\\chi(\\nu(t)) = -\\omega(\\chi(t))\\ , \\quad t\\in Z,\\ \\chi\\in X^{*}(\\Cent)\\ ;\n\\end{equation*}\nnote that $V = \\Hom_{{\\mathbb Z}}(X^{*}(\\Cent), {\\mathbb R})$ since\n$X^{*}(\\Cent) \\subseteq X^{*}(\\Split)$ is a finite index subgroup. \n\nWe now extend $\\nu$ to an action of $N$ by affine transformations. Let $H\n= \\ker\\nu$, which is the maximal compact subgroup of $Z$. Then $Z\/H$ is a\nfree abelian group with rank $= \\dim_{\\mathbb R} V = k_{v}\\text{-rank}\\: \\mathbf G$. The group $W'\n= N\/H$ is an extension of $W$ by $Z\/H$ and there exists an affine action of\n$W'$ on $A$ which makes the following diagram commute\n\\cite{Landvogt}*{1.6}:\n\\begin{equation*}\n\\begin{CD}\n1 @>>> Z\/H @>>> W' @>>> W @>>> 1 \\\\\n@. @VVV @VVV @VVV \\\\\n1 @>>> V @>>> \\Aff(A) @>>> \\mathrm{GL}(V) @>>> 1\\rlap{\\ .}\n\\end{CD}\n\\end{equation*}\nThe action of $W'$ lifts to the desired extension $\\nu\\colon N \\to \\Aff(A)$.\n\nFor each $\\alpha \\in \\Phi$, let $U_{\\alpha}$ be the $k_v$-rational points\nof the connected unipotent subgroup of $\\mathbf G$ which has Lie algebra spanned\nby the root spaces $\\mathfrak g_\\alpha$ and (if $2\\alpha$ is a root)\n$\\mathfrak g_{2\\alpha}$. For $u\\in U_{\\alpha}\\setminus \\{1\\}$, let $m(u)$\nbe the unique element of $N\\cap U_{-\\alpha}uU_{-\\alpha}$\n\\cite{Landvogt}*{0.19}; in $\\SL_2$, for example,\n$m\\left(\\left(\\begin{smallmatrix} 1 & x \\\\ 0\\vphantom{x^{-1}} &\n 1 \\end{smallmatrix}\\right)\\right) = \\left(\\begin{smallmatrix} 0 & x\n \\\\ -x^{-1} & 0 \\end{smallmatrix}\\right)$. The element $m(u) \\in N$ acts\non $A$ by an affine reflection $\\nu(m(u))$ whose associated linear\ntransformation is $r_\\alpha$. The hyperplanes fixed by these affine\nreflections for all $\\alpha$ and $u$ are the \\emph{walls} of $A$. The\nconnected components of the complement of the union of the walls are called\nthe \\emph{chambers} of $A$; since we assume $\\mathbf G$ is almost simple, these\nare (open) simplices. A \\emph{face} of $A$ is an open face of a\nchamber. The affine space $A$ is thus a simplicial complex (with the open\nsimplices being faces) and the action of $N$ is simplicial.\n\nFor convenience we identify $A$ with $V$ by choosing a ``zero'' point $o\\in\nA$. For $\\alpha \\in \\Phi$, define $\\phi_\\alpha\\colon U_{\\alpha} \\to {\\mathbb R}\n\\cup \\{\\infty\\}$ by setting $\\phi_\\alpha(1)=\\infty$ and requiring for\n$u\\neq 1$ that the function $x\\mapsto \\alpha(x) + \\phi_\\alpha(u)$ vanishes\non the wall fixed by $\\nu(m(u))$. For $\\ell \\in {\\mathbb R}$, let\n\\begin{equation*}\nU_{\\alpha,\\ell} = \\{\\, u\\in U_{\\alpha} \\mid \\phi_\\alpha(u) \\ge \\ell\\,\\}\\ .\n\\end{equation*}\nThese are compact open subgroups and define a decreasing exhaustive and\nseparated filtration of $U_{\\alpha}$ which has ``jumps'' only for $\\ell$ in\nthe discrete set $\\phi_\\alpha( U_{\\alpha}\\setminus \\{1\\})$. The affine\nfunction $\\alpha + \\ell$ is called an \\emph{affine root} if for some $u\\in\nU_{\\alpha}\\setminus \\{1\\}$, $\\ell = \\phi_\\alpha(u)$ and (if $2\\alpha$ is a\nroot) $\\phi_\\alpha(u)= \\sup \\phi_\\alpha(u U_{2\\alpha})$; let\n$r_{\\alpha,\\ell} = \\nu(m(u))$ be the corresponding affine reflection. Note\nthat the zero set of an affine root is a wall of\n$A$ and every wall of $A$ arises in this fashion.\n\nDenote the set of affine roots by $\\Phi_{\\mathrm{af}}$; it is an \\emph{affine root\n system} in the sense of \\cite{Macdonald}. The Weyl group $W_{\\mathrm{af}}$ of the\naffine root system $\\Phi_{\\mathrm{af}}$ is the group generated by $r_{\\alpha,\\ell}$\nfor $\\alpha + \\ell \\in \\Phi_{\\mathrm{af}}$; it is an affine Weyl group in the sense\nof \\cite{Bourbaki}*{Ch.~VI, \\S2} associated to a reduced root system (not\nnecessarily $\\Phi$). Since we assume $\\mathbf G$ is simply connected, $W_{\\mathrm{af}} =\n\\nu(N) \\cong W'$.\n\nThe \\emph{apartment} associated to $\\Split$ consists of the affine\nsimplicial space $A$ together with the action of $N$, the affine root\nsystem $\\Phi_{\\mathrm{af}}$, and the filtration of the root groups,\n$(U_{\\alpha,\\ell})_{\\substack{\\alpha\\in\\Phi \\\\ \\ell\\in {\\mathbb R}}}$.\n\n\\subsection{The building}\n\\label{ssectBuilding}\n\nFor $x\\in A$, let $U_x$ be the group generated by $U_{\\alpha,\\ell}$ for all\n$\\alpha + \\ell \\in\\Phi_{\\mathrm{af}}$ such that $(\\alpha + \\ell)(x) \\ge 0$. The\n\\emph{building} of $\\mathbf G$ over $k_v$ is defined \\cite{BruhatTits1}*{(7.4.2)}\nto be\n\\begin{equation*}\nX_v = (G\\times A ) \/ \\!\\sim \\ ,\n\\end{equation*}\nwhere $(gnp,x) \\sim (g, \\nu(n)x)$ for all $n\\in N$ and $p \\in H U_x$. We\nidentify $A$ with the subset of $X_v$ induced by $\\{1\\}\\times A $.\n\nThe building $X_v$ has an action of $\\mathbf G(k_v)$ induced by left\nmultiplication on the first factor of $G\\times A$. Under this action, $N$\nacts on $A\\subset X_v$ via $\\nu$ and $U_{\\alpha,\\ell}$ fixes the points in\nthe half-space of $A$ defined by $\\alpha +\\ell\\ge 0$. The simplicial\nstructure on $A$ induces one on $X_v$ and the action of $\\mathbf G(k_v)$ is\nsimplicial. The subcomplex $gA\\subset X_v$ may be identified with the\napartment corresponding to the maximal split torus $g\\Split g^{-1}$.\n\nChoose an inner product on $V$ which is invariant under the Weyl group $W$;\nthe resulting metric on $A$ may be transferred to any apartment by using the\naction of $\\mathbf G(k_v)$. These metrics fit together to give a well-defined\nmetric on $X_{v}$ which is invariant under $\\mathbf G(k_{v})$\n\\cite{BruhatTits1}*{(7.4.20)} and complete \\cite{BruhatTits1}*{(2.5.12)}.\nGiven two points $x$, $y\\in X_v$, there exists an apartment $gA$ of $X_v$\ncontaining them \\cite{BruhatTits1}*{(7.4.18)}. Since $gA$ is an affine\nspace we can connect $x$ and $y$ with a line segment, $t \\mapsto tx +\n(1-t)y$, $ t \\in [0,1]$; this segment is independent of the choice of\napartment containing the two points and in fact is the unique geodesic\njoining $x$ and $y$.\n\n\\begin{prop}[\\cite{BruhatTits1}*{(7.4.20)}]\nThe mapping $t \\mapsto tx + (1-t)y$ of $[0,1] \\times X_{v} \\times X_{v}\n\\rightarrow X_{v}$ is continuous and thus $X_{v}$ is contractible.\n\\end{prop}\n\nIn fact it follows from \\cite{BruhatTits1}*{(3.2.1)} that $X_v$ is a\n$\\CAT(0)$-space. (Recall that a $\\CAT(0)$-space is a metric space where\nthe distance between any two points is realized by a geodesic and every\ngeodesic triangle is thinner than the corresponding triangle of the same\nside lengths in the Euclidean plane; see \\cite{BH} for a comprehensive\ndiscussion of $\\CAT(0)$-spaces.) Besides affine buildings such as $X_v$,\nanother important class of $\\CAT(0)$-spaces are the simply connected,\nnon-positively curved Riemannian manifolds such as $X_\\infty$.\n\n\\subsection{Stabilizers}\n\\label{ssectStabilizersBuilding}\n\nFor $\\Omega \\subset X_{v}$, let $\\mathbf G(k_{v})_{\\Omega}$ be the subgroup that\nfixes $\\Omega$ pointwise (the \\emph{fixateur} of $\\Omega$). Suppose now\nthat $\\Omega \\subseteq A$ and set \\begin{equation*}\nU_{\\Omega} = \\langle \\, U_{\\alpha,\\ell} \\mid (\\alpha+\\ell)(\\Omega) \\geq 0,\\,\n\\alpha+\\ell\\in \\Phi_{\\mathrm{af}}\\, \\rangle\\ .\n\\end{equation*}\nSince $\\mathbf G$ is simply connected and the valuation $\\omega$ is discrete,\n$\\mathbf G(k_{v})_{\\Omega} = HU_{\\Omega}$ (see \\cite{BruhatTits1}*{(7.1.10),\n (7.4.4)}). In particular, the stabilizer of $x\\in A$ is the compact\nopen subgroup $\\mathbf G(k_{v})_x = HU_x$.\n\nIf $F$ is a face of $A$ and $x\\in F$, then the set of affine roots which\nare nonnegative at $x$ is independent of the choice of $x\\in F$. Thus\n$\\mathbf G(k_{v})_{F} = \\mathbf G(k_{v})_x$. Note that an element of $\\mathbf G(k_v)$ which\nstabilizes $F$ also fixes the barycenter $x_F$ of $F$; thus $\\mathbf G(k_v)_F$ is\nthe stabilizer subgroup of $F$. The stabilizer subgroups for the building\nof $\\SL_2$ (a tree) are calculated in \\cite{SerreTrees}*{II, 1.3}.\n\nLet $\\P$ be a parabolic $k_v$-subgroup which without loss of generality we\nmay assume contains the centralizer of $\\Split$; let $\\mathbf N_{\\P}$ be its\nunipotent radical. Let $\\Phi_P = \\{\\, \\alpha \\in \\Phi \\mid U_\\alpha\n\\subseteq \\mathbf N_{\\P}(k_v) \\, \\}$ and set $E_P = \\{\\, v\\in V \\mid \\alpha(v) \\ge\n0, \\, \\alpha \\in \\Phi_P\\, \\}$; note that $\\Phi_P$ is contained in a positive\nsystem of roots and hence $E_P$ is a cone with nonempty interior.\n\n\\begin{lem}\n\\label{lemUnipotentsHaveFixedPoints}\nFor $u \\in \\mathbf N_{\\P}(k_v)$ there exists $x\\in A$ such that $x + E_P$ is\nfixed pointwise by $u$. In particular, $u$ belongs to a compact open subgroup.\n\\end{lem}\n\n\\begin{proof}\nSince $\\mathbf N_{\\P}(k_v)$ is generated by $(U_\\alpha)_{\\alpha\\in\\Phi_P}$, there\nexists $\\ell\\in {\\mathbb R}$ such that $u$ belongs to the group generated by\n$(U_{\\alpha,\\ell})_{\\alpha\\in\\Phi_P}$. Since $U_{\\alpha,\\ell}$ fixes the\npoints in the half-space of $A$ defined by $\\alpha +\\ell\\ge 0$,\nchoosing $x\\in A$ such that $\\alpha(x) \\ge -\\ell$ for all $\\alpha\\in\n\\Phi_P$ suffices.\n\\end{proof}\n\n\\section{The reductive Borel-Serre and Satake compactifications: the\n $S$-arithmetic case}\n\\label{sectCompactificationsSArithmetic}\n\nWe now consider a general $S$-arithmetic subgroup $\\Gamma$ and define a\ncontractible space $X=X_S$ on which $\\Gamma$ acts properly. If the $k$-rank\nof $\\mathbf G$ is positive, as we shall assume, $\\Gamma\\backslash X$ is noncompact\nand it is important to compactify it. Borel and Serre \\cite{BS2} construct\n$\\ga\\backslash\\osp^{BS}$, the analogue of $\\Gamma\\backslash\\overline{X}_\\infty^{BS}$ from\n\\S\\ref{ssectBSarith}, and use it to study the cohomological finiteness of\n$S$-arithmetic subgroups. In this section we recall their construction and\ndefine several new compactifications of $\\Gamma\\backslash X$ analogous to\nthose in \\S\\ref{sectCompactificationsArithmetic}.\n\n\\subsection{\\boldmath The space $\\Gamma\\backslash X$ associated to an\n$S$-arithmetic group}\n\nLet $S$ be a finite set of places of $k$ containing the infinite places\n$S_\\infty$ and let $S_f = S \\setminus S_\\infty$. Define\n\\begin{equation*}\nG =G_{\\infty}\\times \\prod_{v\\in S_{f}} \\mathbf G(k_{v}),\n\\end{equation*}\nwhich is a locally\ncompact group, and\n\\begin{equation*}\nX =X_{\\infty}\\times \\prod_{v\\in S_{f}} X_{v}\\ ,\n\\end{equation*}\nwhere $X_v$ is the Bruhat-Tits building associated to $\\mathbf G(k_v)$ as\ndescribed in \\S\\ref{sectBruhatTitsBuildings}. If we need to make clear the\ndependence on $S$, we write $X_S$. $X$ is a locally compact\nmetric space under the distance function induced from the factors. Since\neach factor is a $\\CAT(0)$-space and contractible (see\n\\S\\ref{ssectBuilding}), the same is true for $X$.\n\nThe group $G$ acts isometrically on $X$. We view $\\mathbf G(k)\\subset G$ under\nthe diagonal embedding. Any $S$-arithmetic subgroup $\\Gamma \\subset \\mathbf G(k)$ is\na discrete subgroup of $G$ and acts properly on $X$ \\cite{BS2}*{(6.8)}.\nIt is known that the quotient $\\Gamma\\backslash X$ is compact if and\nonly if the $k$-rank of $\\mathbf G$ is equal to 0. In the following, we assume\nthat the $k$-rank of $\\mathbf G$ over $k$ is positive. Then for every $v\\in\nS_{f}$, the $k_{v}$-rank of $\\mathbf G$ is also positive.\n\n\\subsection{The Borel-Serre compactification}\n\\label{ssectBorelSerreSarithmetic}\nDefine\n\\begin{equation*}\n\\overline{X}^{BS} = \\overline{X}_\\infty^{BS}\\times \\prod_{v\\in S_{f}}X_{v} \\ ,\n\\end{equation*}\nwhere $\\overline{X}_\\infty^{BS}$ is as in \\S\\ref{ssectBSarith}. This space\nis contractible and the action of $\\mathbf G(k)$ on $X$ extends to a continuous\naction on $\\overline X^{BS}$. The action of any $S$-arithmetic subgroup\n$\\Gamma$ on $\\overline{X}^{BS}$ is proper \\cite{BS2}*{(6.10)}. When\n$S_f=\\emptyset$ this is proved in \\cite{Borel-Serre} as mentioned in\n\\S\\ref{ssectBSarith}; in general, the argument is by induction on $|S_f|$.\nThe key points are \\cite{BS2}*{(6.8)}:\n\\begin{enumerate}\n\\item The covering of $X_v$ by open stars $V(F)$ about the barycenters\n of faces $F$ satisfies\n\\begin{equation*}\n\\gamma V(F)\\cap V(F) \\neq \\emptyset \\quad \\Longleftrightarrow \\quad\n\\gamma\\in\\Gamma_{F} = \\Gamma\\cap \\mathbf G(k_v)_F \\text{ , and}\n\\end{equation*}\n\\item For any simplex $F \\subset X_{v}$, $\\Gamma_F$ is an\n $(S\\setminus\\{v\\})$-arithmetic subgroup and hence by induction acts\n properly on $\\overline{X}_{S\\setminus\\{v\\}}^{BS}$.\n\\end{enumerate}\nFurthermore $\\Gamma\\backslash \\overline{X}^{BS}$ is compact Hausdorff\n\\cite{BS2}*{(6.10)} which follows inductively from\n\\begin{enumerate}[resume]\n\\item There are only finitely many $\\Gamma$-orbits of simplices in $X_{v}$ for\n $v\\in S_f$ and the quotient of $\\overline{X}^{BS}_\\infty$ by an\n arithmetic subgroup is compact.\n\\end{enumerate}\n\n\\subsection{The reductive Borel-Serre compactification}\n\\label{subsectRBSSarith}\nDefine\n\\begin{equation*}\n\\overline{X}^{RBS}= \\overline{X}_\\infty^{RBS}\\times \\prod_{v\\in S_f}X_{v} \\ .\n\\end{equation*}\nThere is a $\\mathbf G(k)$-equivariant surjection $\\overline{X}^{BS} \\to\n\\overline{X}^{RBS}$ induced from the surjection in \\S\\ref{subsectRBSarith}.\n\\begin{prop}\n\\label{propDiscontinuousRBS}\nAny $S$-arithmetic subgroup $\\Gamma$ of $\\mathbf G(k)$ acts discontinuously on\n$\\overline{X}^{RBS}$ with a compact Hausdorff quotient $\\Gamma\\backslash\\overline{X}^{RBS}$.\n\\end{prop}\nThe proposition is proved similarly to the case of $\\ga\\backslash\\osp^{BS}$ outlined in\n\\S\\ref{ssectBorelSerreSarithmetic}; one replaces ``proper'' by\n``discontinuous'' and begins the induction with Lemma\n~\\ref{lemRBSDiscontinuous}. The space $\\Gamma\\backslash\\overline{X}^{RBS}$ is the\n\\emph{reductive Borel-Serre compactification} of $\\Gamma\\backslash X$.\n\n\\subsection{Satake compactifications}\n\\label{subsectSatakeSArith}\nLet $(\\tau,V)$ be a spherical representation of\n$\\operatorname{Res}_{k\/{\\mathbb Q}}\\mathbf G$ as in \\S\\ref{subsectSatakeArith} and define\n\\begin{equation*}\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}=\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}\n\\times\\prod_{v\\in S_{f}} X_{v}\\ .\n\\end{equation*}\nThere is a $\\mathbf G(k)$-equivariant surjection $\\overline{X}^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ induced by $\\overline{X}_\\infty^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}_\\infty^{\\tau}$ from \\S\\ref{subsectSatakeArith}.\n\n\\begin{prop}\n\\label{propDiscontinuousSatake}\nAssume that the Satake compactification $\\overline{X}_\\infty^{\\tau}$ is\ngeometrically rational. Then any $S$-arithmetic subgroup $\\Gamma$ acts\ndiscontinuously on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ with a compact Hausdorff\nquotient $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$.\n\\end{prop}\n\nThe compact quotient $\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is\ncalled the \\emph{Satake compactification} associated with $(\\tau,V)$.\n\n\\section{The fundamental group of the compactifications and applications to\nthe congruence subgroup kernel}\n\\label{sectFundGrpArithmetic}\n\nIn this section we state our main result, Theorem ~\\ref{thmMainArithmetic},\nwhich calculates the fundamental group of the reductive Borel-Serre and the\nSatake compactifications of $\\Gamma\\backslash X$. We then apply the main result\nto identify the congruence subgroup kernel with certain fundamental groups.\nThe proof of Theorem ~\\ref{thmMainArithmetic} is postponed to\n\\S\\ref{sectProofArithmetic}.\n\nThroughout we fix a spherical representation $(\\tau,V)$ such that\n$\\overline{X}_\\infty^{\\tau}$ is geometrically rational.\n\n\\begin{defi} Let $\\Gamma$ be a group acting continuously on a topological\n space $Y$. For each point $y\\in Y$, let $\\Gamma_{y} =\\{\\,g\\in\\Gamma\\mid\n gy=y\\,\\}$ be the \\emph{stabilizer subgroup} of $y$ in $\\Gamma$. The\n \\emph{fixed subgroup} $\\Gamma_{f}$ is the subgroup generated by the\n stabilizer subgroups $\\Gamma_{y}$ for all $y\\in Y$. (The fixed subgroup is\n obviously normal.)\n\\end{defi}\n\nIn our situation of an $S$-arithmetic subgroup $\\Gamma$ acting on $\\overline{X}^{RBS}$\nand ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$, we denote $\\Gamma_f$ by $\\Gamma_{f,RBS}$ and\n$\\Gamma_{f,\\tau}$ respectively. The main result of this paper is the\nfollowing theorem.\n\n\\begin{thm}\n\\label{thmMainArithmetic}\nFor any $S$-arithmetic subgroup $\\Gamma$, there exists a commutative diagram\n\\begin{equation*}\n\\begin{CD}\n\\pi_{1}(\\ga\\backslash\\oX^{RBS}) @<\\cong<< \\Gamma\/\\Gamma_{f,RBS} \\\\\n@VVV @VVV \\\\\n\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}) @<\\cong<<\n\\Gamma\/\\Gamma_{f,\\tau}\n\\end{CD}\n\\end{equation*}\nwhere the horizontal maps are isomorphisms and the vertical maps are\nsurjections induced by the $\\Gamma$-equivariant projection $\\overline{X}^{RBS} \\to\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ and the inclusion $\\Gamma_{f,RBS} \\subseteq\n\\Gamma_{f,\\tau}$.\n\\end{thm}\n\nThe proof of the theorem will be given in \\S\\ref{sectProofArithmetic}. In\nthe remainder of this section we present some applications to the\ncongruence subgroup kernel. To do this we first need to calculate\n$\\Gamma_{f,RBS}$ and $\\Gamma_{f,\\tau}$ which will require the information on\nstabilizers from \\S\\S\\ref{subsectRBSarith}, \\ref{subsectSatakeArith}, and\n\\ref{ssectStabilizersBuilding}.\n\nLet $\\P$ be a parabolic $k$-subgroup $\\P$ of $\\mathbf G$. The $S$-arithmetic\nsubgroup $\\Gamma$ induces $S$-arithmetic subgroups $\\Gamma_{P}=\\Gamma\\cap\n\\P(k)\\subseteq \\P(k)$, $\\Gamma_{N_{P}} = \\Gamma\\cap \\mathbf N_{\\P}(k) \\subseteq\n\\mathbf N_{\\P}(k)$, and $\\Gamma_{L_{P}} = \\Gamma_{P}\/\\Gamma_{N_{P}} \\subseteq \\mathbf\nL_{\\P}(k)$, as well as $\\Gamma_{P_\\tau} = \\Gamma\\cap \\P_\\tau(k) \\subseteq\n\\P_\\tau(k)$ and $\\Gamma_{H_{P,\\tau}} = \\Gamma_P \/ \\Gamma_{P_\\tau} \\subseteq \\mathbf\nH_{\\P, \\tau}(k)$.\n\nLet $E\\Gamma\\subseteq \\Gamma$ be the subgroup generated by $\\Gamma_{N_{P}}$\nfor every parabolic $k$-subgroup $\\P$ of $\\mathbf G$. Since $\\gamma\n\\mathbf N_{\\P}\\gamma^{-1}=\\mathbf N_{\\gamma \\P\\gamma^{-1}}$ for $\\gamma \\in \\Gamma$,\n$E\\Gamma$ is clearly normal. Let $E_{\\tau}\\Gamma\\subseteq \\Gamma$ be the\nsubgroup generated by $\\Gamma_{P_\\tau} \\cap \\bigcap_{v\\in S_f} K_v$ for\nevery $\\tau$-saturated parabolic $k$-subgroup $\\P$ of $\\mathbf G$ and compact open\nsubgroups $K_v\\subset \\mathbf G(k_v)$. As above, $E_{\\tau}\\Gamma$ is normal.\nSince $\\Gamma_{N_P}$ is generated by $\\Gamma_{N_P} \\cap \\bigcap_{v\\in S_f} K_v$\nfor various $K_v$ by Lemma~\\ref{lemUnipotentsHaveFixedPoints}, it is easy\nto see that $E\\Gamma \\subseteq E_\\tau\\Gamma$.\n\nA subgroup $\\Gamma\\subset \\mathbf G(k)$ is \\emph{neat} if the subgroup of ${\\mathbb C}$\ngenerated by the eigenvalues of $\\rho(\\gamma)$ is torsion-free for any\n$\\gamma\\in\\Gamma$. Here $\\rho$ is a faithful representation $\\mathbf G\\to \\GL_N$\ndefined over $k$ and the condition is independent of the choice of $\\rho$.\nClearly any neat subgroup is torsion-free. Any $S$-arithmetic subgroup has a\nnormal neat subgroup of finite index \\cite{Borel}*{\\S17.6}; the image of a\nneat subgroup by a morphism of algebraic groups is neat\n\\cite{Borel}*{\\S17.3}.\n\n\\begin{prop}\n\\label{propGammaFixedIsEGamma}\nLet $\\Gamma$ be an $S$-arithmetic subgroup. Then $E\\Gamma \\subseteq\n\\Gamma_{f,RBS}$ and $E_{\\tau}\\Gamma \\subseteq \\Gamma_{f,\\tau}$. If $\\Gamma$\nis neat then equality holds for both.\n\\end{prop}\n\n\\begin{proof}\nWe proceed by induction on $\\vert S_{f}\\vert$. Suppose first that\n$S_{f}=\\emptyset$. By Lemma ~\\ref{lemStabilizersRBS}, $\\Gamma_{N_P}$\nstabilizes any point of $X_{P} \\subseteq \\overline{X}^{RBS}_\\infty$ for any\nparabolic $k$-subgroup $\\P$, and hence $E\\Gamma \\subseteq \\Gamma_{f,RBS}$.\nLikewise by Lemma ~\\ref{lemStabilizersSatake}, $\\Gamma_{P_\\tau}$ stabilizes\nany point of $X_{P,\\tau} \\subset {}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ and so\n$E_{\\tau}\\Gamma \\subseteq \\Gamma_{f,\\tau}$.\n\nIf $\\Gamma$ is neat, then $\\Gamma_{L_{P}}$ and $\\Gamma_{H_{P,\\tau}}$ are\nlikewise neat and hence torsion-free. The actions of $\\Gamma_{L_P}$ and\n$\\Gamma_{H_{P,\\tau}}$ are proper and hence $\\Gamma_{L_{P},z}$ and\n$\\Gamma_{H_{P,\\tau},z}$ are finite. Thus these stabilizer subgroups must\nbe trivial. It follows then from Lemmas ~\\ref{lemStabilizersRBS} and\n\\ref{lemStabilizersSatake} that $E\\Gamma = \\Gamma_{f,RBS}$ and\n$E_{\\tau}\\Gamma = \\Gamma_{f,\\tau}$.\n\nNow suppose that $v \\in S_{f}$ and let $S' = S \\setminus\\{v\\}$. Write\n$\\overline{X}^{RBS} = \\overline{X}_{S'}^{RBS} \\times X_v$. Suppose that\n$\\gamma\\in \\Gamma_{N_P}$ for some parabolic $k$-subgroup $\\P$. By Lemma\n~\\ref{lemUnipotentsHaveFixedPoints}, $\\gamma \\in \\mathbf G(k_v)_y$ for some $y\\in\nX_{v}$. Thus $\\gamma \\in \\Gamma' \\cap \\mathbf N_{\\P}(k)$, where $\\Gamma' = \\Gamma\\cap\n\\mathbf G(k_v)_y$. Since $\\mathbf G(k_v)_y$ is a compact open subgroup, $\\Gamma'$ is an\n$S'$-arithmetic subgroup. By induction $\\gamma = \\gamma_1 \\dots \\gamma_m$\nwhere $\\gamma_i\\in\\Gamma'_{x_i}$ with $x_i\\in \\overline{X}_{S'}^{RBS}$. Since\neach $\\gamma_i\\in \\Gamma_{(x_i,y)} \\subset \\Gamma_{f,RBS}$, we see $E\\Gamma\n\\subseteq \\Gamma_{f,RBS}$. The proof that $E_{\\tau}\\Gamma \\subseteq\n\\Gamma_{f,\\tau}$ is similar since if $\\gamma \\in \\Gamma_{P_\\tau} \\cap\n\\bigcap_{v\\in S_f} K_v$ then $\\gamma \\in \\mathbf G(k_v)_y$ for some $y\\in X_v$\n\\cite{BruhatTits1}*{(3.2.4)}.\n\nAssume that $\\Gamma$ is neat. Let $(x,y) \\in \\overline{X}_{S'}^{RBS}\\times\nX_{v}$, and let $F$ be a face of $X_v$ containing $y$. As above,\n$ \\Gamma_F = \\Gamma\\cap \\mathbf G(k_v)_F$ is $S'$-arithmetic and, in this case, neat.\nSo by induction, $\\Gamma_{F,x} \\subseteq E(\\Gamma_F)\\subseteq E\\Gamma$. But\nsince $\\mathbf G(k_{v})_{y}=\\mathbf G(k_{v})_{F}$, $\\Gamma_{(x,y)} =\n\\Gamma_{F,x}$. Therefore $\\Gamma_{f,RBS} \\subseteq E\\Gamma $. A similar\nargument shows that $\\Gamma_{f,\\tau} \\subseteq E_{\\tau}\\Gamma $.\n\\end{proof}\n\nWe now can deduce several corollaries of Theorem ~\\ref{thmMainArithmetic}\nand Proposition ~\\ref{propGammaFixedIsEGamma}.\n\n\\begin{cor}\n\\label{corNeat}\n$\\pi_{1}(\\ga\\backslash\\oX^{RBS})$ is a quotient of $\\Gamma\/ E\\Gamma$ and\n$\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ is a quotient of\n$\\Gamma\/ E_\\tau\\Gamma$. If $\\Gamma$ is neat, then $\\pi_{1}(\\ga\\backslash\\oX^{RBS}) \\cong\n\\Gamma\/ E\\Gamma$ and $\\pi_{1}(\\Gamma\\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}) \\cong \\Gamma\/ E_\\tau\\Gamma$.\n\\end{cor}\n\n\\begin{cor}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, $\\pi_{1}(\\ga\\backslash\\oX^{RBS})$ and\n$\\pi_{1}(\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ are finite.\n\\end{cor}\n\\begin{proof}\nUnder the rank assumptions, $E\\Gamma$ is $S$-arithmetic\n\\citelist{\\cite{Margulis} \\cite{Ra2}*{Theorem~ A, Corollary~ 1}}.\n\\end{proof}\n\n\\begin{cor}\n\\label{corRankTwoAndUp}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, then $C(S,\\mathbf G) =\n\\varprojlim\\limits_{{\\mathfrak a}} \\pi_{1}(\\Gamma({\\mathfrak a})\\backslash\\overline{X}^{RBS})$, where ${\\mathfrak a}$\nranges over nonzero ideals of ${\\mathcal O}$. These fundamental groups and the\nlimit are finite.\n\\end{cor}\n\\begin{proof}\nUnder the rank hypothesis, Raghunathan proves that the congruence kernel is\nthe projective limit of $\\Gamma({\\mathfrak a})\/ E\\Gamma({\\mathfrak a})$ (see\n\\eqref{eqnCongruenceKernel} in \\S\\ref{ssectElementaryMatrices}).\nFurthermore these groups are finite (see the discussion in\n\\S\\S\\ref{ssectKnownResults}, \\ref{ssectElementaryMatrices}). Now apply\nCorollary ~\\ref{corNeat} and the fact that $\\Gamma({\\mathfrak a})$ is neat for ${\\mathfrak a}$\nsufficiently small.\n\\end{proof}\n\nSet $\\Gamma^*({\\mathfrak a}) = \\bigcap_{{\\mathfrak b}\\neq 0} E\\Gamma({\\mathfrak a})\\cdot \\Gamma({\\mathfrak b})$ where ${\\mathfrak b}$ runs\nover nonzero ideals of ${\\mathcal O}$. Clearly\n\\begin{equation}\n\\label{eqnGammaStar}\nE\\Gamma({\\mathfrak a}) \\subseteq \\Gamma^*({\\mathfrak a}) \\subseteq \\Gamma({\\mathfrak a}).\n\\end{equation}\nBy Raghunathan's Main Lemma \\cite{Ra1}*{(1.17)}, for every nonzero ideal\n${\\mathfrak a}$ there exists a nonzero ideal ${\\mathfrak a}'$ such that $\\Gamma^*({\\mathfrak a})\\supseteq\n\\Gamma({\\mathfrak a}')$. Thus $\\Gamma^*({\\mathfrak a})$ is the smallest $S$-congruence subgroup\ncontaining $E\\Gamma({\\mathfrak a})$.\n\n\\begin{cor}\n\\label{corIdentifyCSG}\nIf $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$, then $C(S,\\mathbf G) =\n\\pi_{1}(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS})$ for any sufficiently small nonzero\nideal ${\\mathfrak a}$ of ${\\mathcal O}$.\n\\end{cor}\n\\begin{proof}\nSince $\\Gamma^*({\\mathfrak a})$ is an $S$-congruence subgroup,\nequations \\eqref{eqnCongruenceKernel} and \\eqref{eqnGammaStar} imply that\n\\begin{equation*}\nC(S,\\mathbf G) = \\varprojlim\\limits_{{\\mathfrak a}} \\Gamma({\\mathfrak a})\/ E\\Gamma({\\mathfrak a}) \\cong\n\\varprojlim\\limits_{{\\mathfrak a}} \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a}).\n\\end{equation*}\nSince $C(S,\\mathbf G)$ is finite, the second limit will stabilize if we show\n\\begin{equation*}\n \\Gamma^*({\\mathfrak b})\/ E\\Gamma({\\mathfrak b}) \\longrightarrow\n \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a})\n\\end{equation*}\nis surjective for ${\\mathfrak b}\\subset {\\mathfrak a}$. But this follows from Raghunathan's\nMain Lemma \\cite{Ra1}*{(1.17)} applied to ${\\mathfrak b}$ and the definition of\n$\\Gamma^*({\\mathfrak a})$. Finally we note that that\n$\\pi_{1}(\\Gamma^*({\\mathfrak a})\\backslash\\overline{X}^{RBS}) \\cong \\Gamma^*({\\mathfrak a}) \/ E\\Gamma({\\mathfrak a})$ by\nCorollary ~\\ref{corNeat} and the fact that $E\\Gamma({\\mathfrak a}) = E\\Gamma^*({\\mathfrak a})$ (apply\n$E$ to \\eqref{eqnGammaStar}).\n\\end{proof}\n\n\\begin{rem}\nFrom the point of view of identifying the congruence subgroup kernel $C(S,\n\\mathbf G)$, Corollary ~\\ref{corIdentifyCSG} shows that the reductive Borel-Serre\ncompactification $\\ga\\backslash\\osp^{RBS}$ is the most natural compactification. On\nthe other hand, the Satake compactifications are important as well. In\nparticular, when $X=X_{\\infty}$ is Hermitian, the Baily-Borel\ncompactification is a normal projective variety and has played an important\nrole in algebraic geometry and number theory. In the cases considered in\n\\citelist{\\cite{hk} \\cite{kn} \\cite{hs} \\cite{ge} \\cite{Gro} \\cite{gro2}},\nthe fundamental group of the Baily-Borel compactification is shown to\nvanish. The maximal Satake compactification is also special among the\nfamily of all Satake compactifications and important for various purposes.\nIn the general situation in this paper, the precise relations between\n$C(S,\\mathbf G)$ and $\\pi_{1}(\\Gamma(\\mathfrak a) \\backslash\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ are not clear, even when $\\mathfrak a$ is a\nsufficiently small ideal, aside from the fact that $\\pi_{1}(\\Gamma^*(\\mathfrak\na) \\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau})$ is a quotient of $C(S,\\mathbf G)$ when\nthe $k\\text{-rank}\\: \\mathbf G >0 $ and $S\\text{-rank}\\: \\mathbf G \\ge 2$.\n\\end{rem}\n\n\\section{Proof of the main theorem}\n\\label{sectProofArithmetic}\nIn this section we give the proof of Theorem ~\\ref{thmMainArithmetic}. The\nmain tool is Proposition ~\\ref{propGrosche}. Part ~\\ref{itemGrosche} in\nthe proposition is used for the proof of the case where $\\Gamma$ is neat; it\nrequires the notion of an \\emph{admissible} map (Definition\n~\\ref{defiAdmissible}). Part ~\\ref{itemArmstrong} is needed in addition to\ncomplete the general case. In order to apply Proposition\n~\\ref{propGrosche} we must first verify that the spaces\n$\\overline{X}^{RBS}$ and ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are simply connected\n(Proposition ~\\ref{propSimply}) and that the $\\Gamma$-actions are admissible\nin the neat case (Proposition ~\\ref{propAdmissibleNeatCase}). Both of\nthese arguments depend on deforming paths to the boundary where the\ngeometry is simpler; this technique is formalized in Lemma\n~\\ref{lemAdmissibilityViaRetract}.\n\nHomotopy of paths $\\omega$ and $\\eta$ will always mean homotopy relative to\nthe endpoints and will be denoted $\\omega \\cong \\eta$. An action of a\ntopological group $\\Gamma$ on a topological space $Y$ will always be a\ncontinuous action.\n\n\\begin{defi}\n\\label{defiAdmissible}\nA continuous surjection $p\\colon Y \\to X$ of topological spaces is\n\\emph{admissible} if for any path $\\omega$ in $X$ with initial point $x_0$\nand final point $x_1$\nand for any $y_0\\in p^{-1}(x_0)$, there exists a path $\\tilde{\\omega}$ in\n$Y$ starting at $y_0$ and ending at $y_1\\in p^{-1}(x_1)$ such that $p\\circ \\tilde \\omega$ is homotopic to\n$\\omega$ relative to the endpoints.w An action of a group $\\Gamma$\non a topological space $Y$ is \\emph{admissible} if the quotient map $Y\\to\n\\Gamma\\backslash Y$ is admissible.\n\\end{defi}\n\n\\begin{prop}\n\\label{propGrosche}\nLet $Y$ be a simply connected topological space and $\\Gamma$ a discrete group\nacting on $Y$. Assume that either\n\\begin{enumerate}\n\\item\\label{itemGrosche} the $\\Gamma$-action is discontinuous and admissible,\n or that\n\\item\\label{itemArmstrong} the $\\Gamma$-action is proper and $Y$ is a locally\n compact metric space.\n\\end{enumerate}\nThen the natural morphism $\\Gamma \\to \\pi_{1}(\\Gamma\\backslash Y)$ induces an\nisomorphism $\\Gamma\/\\Gamma_{f} \\cong \\pi_{1}(\\Gamma\\backslash Y)$.\n\\end{prop}\n\\begin{proof}\nSee \\cite{Gro}*{Satz~5} and \\cite{Armstrong} for hypotheses\n\\ref{itemGrosche} and \\ref{itemArmstrong} respectively .\n\\end{proof}\n\n\\begin{prop}\n\\label{propAdmissibilityImpliesSC}\nLet $p\\colon Y \\to X$ be an admissible continuous map of a simply\nconnected topological space $Y$ and assume that $p^{-1}(x_0)$ is\npath-connected for some $x_0\\in X$. Then $X$ is simply connected.\n\\end{prop}\n\\begin{proof}\nLet $\\omega\\colon [0,1] \\to X$ be a loop based at $x_0$ and let\n$\\tilde\\omega$ be a path in $Y$ such that $p\\circ \\tilde\\omega \\cong\n\\omega$ (relative to the basepoint). Let $\\eta$ be a path in\n$p^{-1}(x_0)$ from $\\tilde\\omega(1)$ to $\\tilde\\omega(0)$. Then the\nproduct $\\tilde\\omega\\cdot \\eta$ is a loop in the simply connected space\n$Y$ and hence is null-homotopic. It follows that $\\omega\\cong p\\circ\n\\tilde \\omega\\cong p\\circ(\\tilde\\omega\\cdot \\eta)$ is null-homotopic.\n\\end{proof}\n\n\\begin{lem}\n\\label{lemAdmissibilityIsLocal}\nA continuous surjection $p\\colon Y \\to X$ of topological spaces is\nadmissible if and only if $X$ can be covered by open subsets $U$\nsuch that $p|_{p^{-1}(U)}\\colon p^{-1}(U) \\to U$ is\nadmissible.\n\\end{lem}\n\\begin{proof}\nBy the Lebesgue covering lemma, any path $\\omega\\colon [0,1] \\to X$ is\nhomotopic to the product of finitely many paths, each of which maps into\none of the subsets $U$. The lemma easily follows.\n\\end{proof}\n\n\\begin{lem}\n\\label{lemAdmissibilityViaRetract}\nLet $p\\colon Y \\to X$ be a continuous surjection of topological spaces.\nAssume there exist deformation retractions $r_t$ of $X$ onto a subspace\n$X_0$ and $\\tilde r_t$ of $Y$ onto $Y_0 = p^{-1}(X_0)$ such that $p\\circ\n\\tilde r_t = r_t \\circ p$. Also assume for all $x\\in X$ that\n$\\pi_0(p^{-1}(x)) \\xrightarrow{\\tilde r_{0*}} \\pi_0(p^{-1}(r_0(x)))$ is\nsurjective. Then $p$ is admissible if and only if $p|_{Y_0}\\colon Y_0\\to\nX_0$ is admissible.\n\\end{lem}\n\n\\setlength{\\pinch}{.002128769252056923\\textwidth}\n\\setlength{\\mim}{2.85427559055181102\\pinch}\n\\begin{figure}[h]\n\\begin{equation*}\n\\begin{xy}\n<0\\mim,-15\\mim>;<3\\mim,-15\\mim>:\n<24\\mim,-3\\mim>=\"c\"+<0\\mim,6\\mim>=\"cdmid\"+<0\\mim,6\\mim>=\"d\",\n(15,0)=\"adown\";(0,10)=\"bdown\" **[bordergrey]\\crv{(10,5)&(5,8)}?(.3)=\"xdown\",\n?(.25)=\"main3\",?(.6)=\"main6\",\n\"adown\"+\"c\";{\"bdown\"+\"c\"} **[verylightgrey]\\crv{ (10,5)+\"c\" & (5,8)+\"c\"},\n(15,10)=\"aup\";(0,20)=\"bup\" **[bordergrey]\\crv{(10,15)&(5,18)}\n?(.5)=\"xup\",\n\"aup\"+\"d\";{\"bup\"+\"d\"} **[bordergrey]\\crv{\"d\"+(10,15) & \"d\"+(5,18)},\n\"adown\",\\blownupslice{bordergrey}{bordergrey},\n\"bdown\",\\blownupslice{verylightgrey}{bordergrey},\n\"main6\",\\blownupslice{verylightgrey}{verylightgrey},\n\"bot\";p+<0\\mim,42\\mim>**\\dir{}?(.65)*\\dir{*}=\"y1\"*+!L{_{y_1}}=\"f3\",\n?(.25)*\\cir<1\\pinch>{}*\\frm{*}=\"f7\",?(.85)*\\cir<1\\pinch>{}*\\frm{*}=\"f1\",\n?(.75)*+{}=\"f2\",?(.35)*+{}=\"f6\",?(.45)*\\cir<1\\pinch>{}*\\frm{*}=\"f5\",\n?(.55)*+{}=\"f4\",?(.05)*+{}=\"f9\",\n\"top\"+<0\\mim,6\\mim>*++!DC\\txt<20\\mim>\\tiny{$ p^{-1}(x_1)$ (marked by\n $\\scriptscriptstyle \\bullet$)\\\\$\\downarrow$},\n\"main6\";\"upper\" **[lightgrey]\\dir{-}?(.5)=\"r0y1\"*\\dir{*}*+!RD{_{\\tilde\n r_0(y_1)}},\n?(.25)=\"eta1\"*\\dir{*}*+!UR{_{\\eta(1)}},\n\"lower\"-<0\\mim,6\\mim>*++!UC\\txt<20\\mim>\\tiny{$\\uparrow$\\\\$p^{-1}(r_0(x_1))$},\n;\"lower\"**[lightgrey]\\dir{--},\n\"upper\";\"upper\"+<0\\mim,6\\mim>**[lightgrey]\\dir{--},\n\"main3\",\\blownupslice{verylightgrey}{verylightgrey},\n\"main3\";\"bot\" **\\crv{~*\\dir{} \"ccp\"},?(.8)=\"x0bot\";p+<0\\mim,37.5\\mim>=\"x0top\"\n**\\dir{}?(.4)=\"y0\"*\\dir{*}*+!L{_{y_0}},\n\"main3\";\"upper\" **\\dir{}?(.35)=\"r0y0\"*\\dir{*}*+!UR{_{\\tilde r_0(y_0)}},\n;\"y0\" **\\dir{},?(.5)+\/u2.25\\pinch\/=\"mcp\",\n\"r0y0\";\"y0\" **\\crv{ \"mcp\"},?(.5)*+!U{_{\\tilde\\sigma_0}},*\\dir{>},\n\"r0y0\"+<0\\pinch,.5\\pinch>;\"y0\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"r0y0\";\"eta1\" **\\dir{},?(.45)+\/r3.75\\pinch\/=\"cp1\",?(.55)+\/l2\\pinch\/=\"cp2\",\n\"eta1\" **\\crv{ \"cp1\" & \"cp2\" },?(.5)*+!UR{_\\eta},*\\dir{>},\n\"r0y0\"+<.35\\pinch,.35\\pinch>;\"eta1\"+<.35\\pinch,.35\\pinch> **\\crv{ \"cp1\"+<.35\\pinch,.35\\pinch> & \"cp2\"+<.35\\pinch,.35\\pinch> },?(.5)*\\dir{>},\n\"eta1\";\"r0y1\" **\\dir{-},?(.5)*+!R{_{\\psi}},*\\dir{>},\n\"eta1\"+<.5\\pinch,0\\pinch>;\"r0y1\"+<.5\\pinch,0\\pinch> **\\dir{-},?(.5)*\\dir{>},\n\"r0y1\";\"y1\" **\\dir{},?(.5)+\/d4.5\\pinch\/=\"mcp\",\n\"r0y1\";\"y1\" **\\crv{ \"mcp\"},?(.5)*+!U{_{\\tilde\\sigma_1}},*\\dir{>},\n\"r0y1\"+<0\\pinch,.5\\pinch>;\"y1\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"adown\"+<-2\\mim,-5\\mim>*{_{Y_0\\quad\\qquad\\subseteq \\quad\\qquad Y}},\n{<79\\mim,15\\mim> \\ar _{p} @{>} <89\\mim,15\\mim>},\n<90\\mim,0\\mim>;<93\\mim,0\\mim>:\n<24\\mim,-3\\mim>=\"c\"+<0\\mim,6\\mim>=\"cdmid\"+<0\\mim,6\\mim>=\"d\",\n(15,0)=\"a\";(0,10)=\"b\" **[bordergrey]\\crv{(10,5)&(5,8)},\n?(.25)=\"main3\"*\\dir{*}*+!UR{_{r_0(x_0)}},?(.6)=\"main6\"*\\dir{*}*+!UR{_{r_0(x_1)}},,\n\"main3\",\\slice{verylightgrey}{verylightgrey},\n\"main3\";\"bot\" **\\crv{~*\\dir{} \"ccp\"},?(.8)=\"x0\"*\\dir{*}*+!U{_{x_0}},\n\"x0\" **\\crv{ \"ccp\"},?(.5)*+!U{_{\\sigma_0}},*\\dir{>},\n\"main3\"+<0\\pinch,.5\\pinch>;\"x0\"+<0\\pinch,.5\\pinch> **\\crv{ \"ccp\"+<0\\pinch,.5\\pinch>},?(.5)*\\dir{>},\n\"main6\",\\slice{verylightgrey}{verylightgrey},\n\"main6\"+\"cdmid\"-<1.5\\mim,0\\mim>=\"x1\"*\\dir{*}*+!DR{_{x_1}}, \n\"a\"+\"cdmid\";{\"b\"+\"cdmid\"} **\\crv{~*\\dir{} \"cdmid\"+(10,5) & \"cdmid\"+(5,8)},\n\"a\"+\"c\";{\"b\"+\"c\"} **[verylightgrey]\\crv{\"c\"+(10,5) & \"c\"+(5,8)},\n\"main6\";\"x1\" **\\dir{},?(.5)+\/d3\\pinch\/=\"mcp\",\n\"main6\";\"x1\" **\\crv{ \"mcp\"},?(.6)*+!U{_{\\sigma_1}},*\\dir{>},\n\"main6\"+<0\\pinch,.5\\pinch>;\"x1\"+<0\\pinch,.5\\pinch> **\\crv{ \"mcp\"+<0\\pinch,.5\\pinch>},?(.6)*\\dir{>},\n\"x0\";\"x1\" **\\dir{},?(.45)+\/r37.5\\pinch\/=\"cp1\",?(.55)+\/l27.5\\pinch\/=\"cp2\",\n\"x1\" **\\crv{ \"cp1\" & \"cp2\" },?(.5)*+!LD{_\\omega},*\\dir{>},\n\"x0\"+<.35\\pinch,.35\\pinch>;\"x1\"+<.35\\pinch,.35\\pinch> **\\crv{ \"cp1\"+<.35\\pinch,.35\\pinch> & \"cp2\"+<.35\\pinch,.35\\pinch> },?(.5)*\\dir{>},\n\"main3\";\"main6\" **\\crv{(10.2,4.4)}?(.6)*\\dir{>}, \n*+!UR{_{r_0\\circ\\omega}},\n\"main3\"+<.35\\pinch,.35\\pinch>;\"main6\"+<.35\\pinch,.35\\pinch> **\\crv{(10.2,4.4)+<.35\\pinch,.35\\pinch>}?(.6)*\\dir{>}, \n\"a\",\\slice{bordergrey}{bordergrey},\n\"b\",\\slice{verylightgrey}{bordergrey},\n\"a\"+\"d\";{\"b\"+\"d\"} **[bordergrey]\\crv{\"d\"+(10,5) & \"d\"+(5,8)},\n\"a\"+<10\\mim,-6\\mim>*{_{X_0\\quad\\subseteq \\quad X}}\n\\end{xy}\n\\end{equation*}\n\\caption{$p\\colon Y \\to X$ as in Lemma~\\ref{lemAdmissibilityViaRetract}}\n\\label{figAdmissibility}\n\\end{figure}\n\n\\begin{proof}\n(See Figure ~\\ref{figAdmissibility}.) Assume $p|_{Y_0}$ is admissible. If\n $\\omega$ is a path in $X$ from $x_0$ to $x_1$, then $\\omega \\cong\n \\sigma_0^{-1} \\cdot (r_0 \\circ \\omega) \\cdot \\sigma_1$ where $\\sigma_i(t)\n = r_t(x_i)$ for $i=0$, $1$. Pick $y_0\\in p^{-1}(x_0)$ and let $\\eta(t)$\n be a path in $Y_0$ starting at $\\tilde r_0(y_0)$ such that $p\\circ \\eta\n \\cong r_0\\circ \\omega$. By assumption there exists $y_1\\in p^{-1}(x_1)$\n such that $\\tilde r_0(y_1)$ is in the same path-component of\n $p^{-1}(r_0(x_1))$ as $\\eta(1)$; let $\\psi$ be any path in\n $p^{-1}(r_0(x_1))$ from $\\eta(1)$ to $\\tilde r_0(y_1)$. Set\n $\\tilde\\omega = \\tilde\\sigma_0^{-1} \\cdot \\eta \\cdot \\psi \\cdot\n \\tilde\\sigma_1$, where $\\tilde \\sigma_i(t) = \\tilde r_t(y_i)$. Then\n $p\\circ \\tilde\\omega \\cong \\sigma_0^{-1} \\cdot (r_0 \\circ \\omega) \\cdot\n \\sigma_1$ and thus $p$ is admissible.\n\\end{proof}\n\nRecall the $\\mathbf G(k)$-equivariant quotient maps $\\overline{X}^{BS}\n\\xrightarrow{p_1} \\overline{X}^{RBS} \\xrightarrow{p_2}\n {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ from \\S\\S\\ref{subsectRBSSarith},\n \\ref{subsectSatakeSArith}.\n\n\\begin{prop}\n\\label{propSimply}\nThe spaces $\\overline{X}^{RBS}$ and ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are\nsimply connected.\n\\end{prop}\n\\begin{proof}\nFor any finite place $v$, the building $X_{v}$ is contractible. So we need\nonly prove that $\\overline{X}^{RBS}_\\infty$ and\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ are simply connected (the case that\n$S_{f} = \\emptyset$). By Proposition ~\\ref{propAdmissibilityImpliesSC},\nLemma ~\\ref{lemAdmissibilityIsLocal}, and the fact that\n$\\overline{X}^{BS}_\\infty$ is contractible, it suffices to find a cover of\n$\\overline{X}^{RBS}_\\infty$ by open subsets $U$ over which $p_1$\n(resp. $p_2\\circ p_1$) is admissible.\n\nConsider first $\\overline{X}^{RBS}_\\infty$. The inverse image\n$p_1^{-1}(X_Q)$ of a stratum $X_Q \\subseteq \\overline{X}^{RBS}_\\infty$ is\n$e(Q) = N_Q\\times X_Q \\subseteq \\overline{X}^{BS}_\\infty$. Set $\\tilde U =\n\\overline A_Q(1) \\times N_Q \\times X_Q \\subseteq \\overline{X}^{BS}_\\infty$\n(compare \\eqref{Pcorner}) and $U=p_1(\\tilde U)$, a neighborhood of $X_Q$; note $p_1^{-1}(U)= \\tilde\nU$. Define a deformation retraction of \n$\\tilde U$ onto\n$e(Q)$ by\n\\begin{equation*}\n\\tilde r_t(a,u,z) =\n\\begin{cases}\n(\\exp(\\frac{1}{t}\\log a), u, z) & \\text{for $t\\in (0,1]$,} \\\\\n(o_Q, u, z) & \\text{for $t=0$.}\n\\end{cases}\n\\end{equation*}\nThis descends to a deformation retraction $r_t$ of $U$ onto $X_Q$. Since\n$p_1|_{e(Q)}\\colon N_Q\\times X_Q \\to X_Q$ is admissible and $N_Q$ is\npath-connected, Lemma ~\\ref{lemAdmissibilityViaRetract} shows that\n$p_1|_{\\tilde U}$ is admissible.\n\nNow consider ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ and a stratum $X_{Q,\\tau}$,\nwhere $\\mathbf Q$ is $\\tau$-saturated. The inverse image $(p_2\\circ\np_1)^{-1}(X_{Q,\\tau})$ is $\\coprod_{\\P^\\dag = \\mathbf Q} e(P) \\subseteq\n\\overline{X}^{BS}_\\infty$; it is an open subset of the closed stratum\n$\\overline{e(Q)} = \\coprod_{\\P \\subseteq \\mathbf Q} e(P)$. For each $\\P$\nsuch that $\\P^\\dag = \\mathbf Q$, we can write $e(P) = N_P\\times X_P = N_P\n\\times X_{Q,\\tau} \\times W_{P,\\tau}$ by \\eqref{eqnBoundaryDecomposition}.\nThus $(p_2\\circ p_1)^{-1}(X_{Q,\\tau}) = Z_Q\\times X_{Q,\\tau}$, where $Z_Q =\n\\coprod_{\\P^\\dag = \\mathbf Q} ( N_P \\times W_{P,\\tau})$. Note that\n$N_Q\\times W_{Q,\\tau}$ is dense in $Z_Q$, so $Z_Q$ is path-connected.\n\nFor $X_{Q,\\tau}\\subset {}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$, the construction\nof $\\tilde U$ is more subtle than in the case of\n$\\overline{X}^{RBS}_\\infty$. The theory of tilings \\cite{sap1}*{Theorem\n ~8.1} describes a neighborhood in $\\overline{X}^{BS}_\\infty$ of the\nclosed stratum $\\overline{e(Q)}$ which is piecewise-analytically\ndiffeomorphic to $\\overline A_Q(1)\\times \\overline{e(Q)}$. (Note however\nthat the induced decomposition on the part of this neighborhood in\n$X_\\infty(Q)$ does \\emph{not} in general agree with that of\n\\eqref{Pcorner}.) We thus obtain a neighborhood $\\tilde U$ of $(p_2\\circ\np_1)^{-1}(X_{Q,\\tau}) = Z_Q \\times X_{Q,\\tau}$ in $\\overline{X}^{BS}_\\infty$ and a\npiecewise-analytic diffeomorphism $\\tilde U \\cong \\overline A_Q(1)\\times\nZ_Q \\times X_{Q,\\tau}$; let $U = p_2\\circ p_1(\\tilde U)$ and note \n$(p_2\\circ p_1)^{-1}(U) = \\tilde U$. Since $Z_Q$ is\npath-connected, we proceed as in the $\\overline{X}^{RBS}_\\infty$ case.\n\\end{proof}\n\n\\begin{rem}\nIt is proved in \\cite{ji2} that every Satake compactification\n$\\overline{X}^{\\tau}_\\infty$ of a symmetric space $X_\\infty$ is a topological ball\nand hence contractible. Though the partial Satake compactification\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is contained in\n$\\overline{X}^{\\tau}_\\infty$ as a subset, their topologies are different and\nthis inclusion is not a topological embedding. Hence, it does not follow\nthat ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is contractible or that a path in\n${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ can be retracted into the interior. In\nfact, it is not known if ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$ is weakly\ncontractible.\n\\end{rem}\n\n\\begin{prop}\\label{propAdmissibleNeatCase}\nFor any neat $S$-arithmetic subgroup $\\Gamma$, the action of $\\Gamma$ on\n$\\overline{X}^{RBS}$ and on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is admissible.\n\\end{prop}\n\n\\begin{proof}\nLet $Y = \\overline{X}^{RBS}$ or ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ and let\n$p\\colon Y \\to \\Gamma\\backslash Y$ be the quotient map, which in this case is\nopen. It suffices to find for any point $x\\in Y$ an open neighborhood $U$\nsuch that $p|_U$ is admissible. For then $p|_{\\Gamma U}$\nis admissible and hence, by Lemma ~\\ref{lemAdmissibilityIsLocal}, $p$ is\nadmissible.\n\nWe proceed by induction on $\\vert S_{f}\\vert$ and we suppose first that\n$S_{f}=\\emptyset$.\n\nSuppose $x$ belongs to the stratum $X_Q$ of $\\overline{X}^{RBS}_\\infty$. Since\n$\\Gamma$ is neat, $\\Gamma_{L_Q}$ is torsion-free. Thus we can choose a\nrelatively compact neighborhood $O_Q$ of $x$ in $X_Q$ so that\n$p|_{O_Q}\\colon O_Q \\to p(O_Q)$ is a homeomorphism. Let $U =\np_1(\\overline A_Q(s) \\times N_Q \\times O_Q) \\subseteq \\overline{X}^{RBS}_\\infty$\nwhere $s>0$; this is a smaller version of the set $U$ constructed in the\nproof of Proposition ~\\ref{propSimply}. By reduction theory, we can choose\n$s$ sufficiently large so that the identifications induced by $\\Gamma$ on $U$\nagree with those induced by $\\Gamma_Q$ \\cite{Zu3}*{(1.5)}. Since $\\Gamma_Q\n\\subseteq N_Q \\widetilde M_Q $, it acts only on the last two factors of\n$\\overline A_Q \\times N_Q \\times X_Q$. Thus the deformation retraction\n$r_t$ of $U$ onto $O_Q$ (from the proof of Proposition ~\\ref{propSimply})\ndescends to a deformation retraction of $p(U)$ onto $p(O_Q)=O_Q$.\nNow apply Lemma ~\\ref{lemAdmissibilityViaRetract} to see that $p|_U$ is\nadmissible.\n\nFor $x$ in the stratum $X_{Q,\\tau}$ of ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}_\\infty$, we\nagain emulate the construction of $U$ from the proof of Proposition\n~\\ref{propSimply}. Specifically let $U= (p_2\\circ p_1)(\\overline\nA_Q(s)\\times Z_Q \\times O_{Q,\\tau})$ where $O_{Q,\\tau}$ is a relatively\ncompact neighborhood of $x$ in $X_{Q,\\tau}$ such that\n$p|_{O_{Q,\\tau}}\\colon O_{Q,\\tau} \\to p(O_{Q,\\tau})$ is a homeomorphism;\nsuch a $O_{Q,\\tau}$ exists since $\\Gamma_{H_{Q,\\tau}}$ is neat and hence\ntorsion-free. By \\cite{sap1}*{Theorem ~8.1}, the identifications induced\nby $\\Gamma$ on $U$ agree with those induced by $\\Gamma_Q$ and these are\nindependent of the $\\overline A_Q(s)$ coordinate. Thus the deformation\nretraction $r_t$ descends to $p(U)$ and we proceed as above.\n\nNow suppose that $v \\in S_{f}$ and let $S' = S \\setminus\\{v\\}$. We\nconsider $Y = \\overline{X}^{RBS}$ which we write as\n$\\overline{X}^{RBS}_{S'}\\times X_{v}$; the case $Y =\n{}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is identical. Following \\cite{BS2}*{(6.8)}, for\neach face $F$ of $X_{v}$ let $x_{F}$ be the barycenter of $F$ and let\n$V(F)$ be the open star of $x_{F}$ in the barycentric subdivision of\n$X_{v}$. The sets $V(F)$ form an open cover of $X_{v}$. For any $\\gamma\n\\in \\Gamma$, $\\gamma V(F) = V(\\gamma F)$. If $F_{1} \\neq F_{2}$ are two\nfaces with $\\dim F_{1} = \\dim F_{2}$, then $V(F_{1}) \\cap V(F_{2}) =\n\\emptyset$. It follows that\n\\begin{equation*}\n\\gamma V(F)\\cap V(F) \\neq \\emptyset \\quad \\Longleftrightarrow \\quad\n\\gamma\\in\\Gamma_{F}\\ ,\n\\end{equation*}\nwhere $\\Gamma_F = \\Gamma \\cap \\mathbf G(k_v)_F$. It follows from\n\\S\\ref{ssectStabilizersBuilding} that $\\Gamma_F$ fixes $F$ pointwise (since\n$\\mathbf G(k_v)_F$ does) and is a neat $S'$-arithmetic subgroup (since $\\mathbf G(k_v)_F$ is\na compact open subgroup of $\\mathbf G(k_v)$)\n\nLet $U = \\overline{X}^{RBS}_{S'}\\times V(F)$ for some open face $F$ of\n$X_{v}$. Define a deformation retraction $r_t$ of $U$ onto\n$\\overline{X}^{RBS}_{S'}\\times F$ by $r_t(w,z) = (w, tz + (1-z)r_F(z))$,\nwhere $r_F(z)$ is the unique point in $F$ which is closest to $z\\in V(F)$.\nThe map $r_t$ is $\\Gamma_F$-equivariant since $\\Gamma_{F}$ fixes $F$\npointwise and acts by isometries. So $r_{t}$ descends to a deformation\nretraction of $p(U)$ onto $(\\Gamma_F\\backslash\n\\overline{X}^{RBS}_{S'})\\times F$. The remaining hypothesis of Lemma\n~\\ref{lemAdmissibilityViaRetract} is satisfied since $r_0(\\gamma w, \\gamma\nz) = r_0(\\gamma w,z)$ for $\\gamma \\in \\Gamma_F$. Since\n$\\overline{X}^{RBS}_{S'}\\times F \\to (\\Gamma_F\\backslash\n\\overline{X}^{RBS}_{S'})\\times F$ is admissible by induction, the lemma\nimplies that $p|_U$ is admissible.\n\\end{proof}\n\nTheorem ~\\ref{thmMainArithmetic} holds if $\\Gamma$ is neat by combining \nPropositions ~\\ref{propGrosche}\\ref{itemGrosche},\n~\\ref{propDiscontinuousRBS}, ~\\ref{propDiscontinuousSatake},\n\\ref{propSimply}, and \\ref{propAdmissibleNeatCase}.\n\n\\begin{cor}\n\\label{corAdmissibleSubgroupNeatCase}\nFor any neat $S$-arithmetic subgroup $\\Gamma$, the actions of $E\\Gamma$ on\n$\\overline{X}^{RBS}$ and $E_\\tau\\Gamma$ on ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ are admissible.\n\\end{cor}\n\\begin{proof}\nBy Proposition ~\\ref{propGammaFixedIsEGamma} the action of $\\Gamma\/E\\Gamma$ on\n$E\\Gamma \\backslash \\overline{X}^{RBS}$ is free and by Proposition\n~\\ref{propDiscontinuousRBS} it is discontinuous. It follows that $E\\Gamma\n\\backslash \\overline{X}^{RBS} \\to (\\Gamma\/E\\Gamma)\\backslash (E\\Gamma\\backslash\n\\overline{X}^{RBS}) = \\Gamma \\backslash \\overline{X}^{RBS}$ is a covering\nspace (in fact a regular covering space) and thus $E\\Gamma $ acts admissibly\nif and only if $\\Gamma$ acts admissibly.\nNow apply the proposition. The case of ${}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ is\ntreated similarly.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem ~\\textup{\\ref{thmMainArithmetic}}]\nLet $\\Gamma'\\subseteq \\Gamma$ be a normal neat subgroup of finite index. The\nidea in the general case is to factor $\\overline{X}^{RBS}\\to \\Gamma\\backslash\n\\overline{X}^{RBS}$ as\n\\begin{equation*}\n\\overline{X}^{RBS}\\to E\\Gamma'\\backslash \\overline{X}^{RBS} \\to\n(\\Gamma\/E\\Gamma')\\backslash (E\\Gamma'\\backslash \\overline{X}^{RBS}) = \\Gamma\\backslash\n\\overline{X}^{RBS}\n\\end{equation*}\nand apply Proposition ~\n\\ref{propGrosche}\\ref{itemGrosche} to the first map and Proposition ~\n\\ref{propGrosche}\\ref{itemArmstrong} to the second map.\n\nBy Proposition ~\\ref{propGammaFixedIsEGamma}, $\\Gamma'_{f,RBS} = E\\Gamma'$ and\nhence $(E\\Gamma')_{f,RBS} = E\\Gamma'$. Thus $E\\Gamma' \\backslash\n\\overline{X}^{RBS}$ is simply connected by Propositions\n~\\ref{propDiscontinuousRBS}, \\ref{propSimply},\n\\ref{propGrosche}\\ref{itemGrosche}, and Corollary\n\\ref{corAdmissibleSubgroupNeatCase}. We now claim that $E\\Gamma' \\backslash\n\\overline{X}^{RBS}$ is locally compact. To see this, note that $E\\Gamma'\n\\backslash \\overline{X}^{BS}$ is locally compact since it is triangulable\n\\cite{BS2}*{(6.10)}. Furthermore the fibers of $p_1'\\colon E\\Gamma'\n\\backslash \\overline{X}^{BS} \\to E\\Gamma' \\backslash \\overline{X}^{RBS}$ have\nthe form $\\Gamma'_{N_P}\\backslash N_P$ which are compact. The claim follows.\nWe can now apply Proposition ~\\ref{propGrosche}\\ref{itemArmstrong} to\n$\\Gamma\\backslash \\overline{X}^{RBS} = (\\Gamma\/E\\Gamma' )\\backslash (E\\Gamma'\n\\backslash\\overline{X}^{RBS})$ and find that $\\pi_1(\\Gamma\\backslash\n\\overline{X}^{RBS}) \\cong (\\Gamma\/E\\Gamma' ) \/ (\\Gamma\/E\\Gamma' )_{f,RBS} \\cong \\Gamma \/\n\\Gamma_{f,RBS}$ as desired. Furthermore the proof shows that the isomorphism\nis induced by the natural morphism $\\Gamma \\to \\pi_1(\\Gamma\\backslash\n\\overline{X}^{RBS})$.\n\nA similar proof using $E_\\tau\\Gamma'$ instead of $E\\Gamma'$ treats the case of\n$\\Gamma\\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$; one only needs to observe\nthat the fibers of $p_2'\\colon E_\\tau\\Gamma' \\backslash \\overline{X}^{RBS} \\to\nE_\\tau\\Gamma' \\backslash {}_{{\\mathbb Q}}\\overline{X}^{\\tau}$ have the form\n$\\Gamma'_{L_{P,\\tau}} \\backslash \\overline{W}_{P,\\tau}^{RBS}$ which are\ncompact.\n\\end{proof}\n\n\\begin{bibdiv}\n\\begin{biblist}\n\\bib{Armstrong}{article}{\n author={Armstrong, M. A.},\n title={The fundamental group of the orbit space of a discontinuous group},\n journal={Proc. Cambridge Philos. Soc.},\n volume={64},\n date={1968},\n pages={299--301},\n}\n\\bib{BB}{article}{\n author={Baily, W. 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Math.},\n volume={116},\n date={1994},\n pages={139--213},\n issn={0020-9910},\n}\n\\bib{Gro}{article}{\n author={Grosche, J{\\\"u}rgen},\n title={\\\"Uber die Fundamentalgruppen von Quotientenr\\\"aumen Siegelscher\n Modulgruppen},\n journal={J. Reine Angew. Math.},\n volume={281},\n date={1976},\n pages={53--79},\n issn={0075-4102},\n}\n\\bib{gro2}{article}{\n author={Grosche, J{\\\"u}rgen},\n title={\\\"Uber die Fundamentalgruppen von Quotientenr\\\"aumen Siegelscher\n und Hilbert-Siegelscher Modulgruppen},\n journal={Nachr. Akad. Wiss. G\\\"ottingen Math.-Phys. Kl. II},\n date={1976},\n number={9},\n pages={119--142},\n issn={0065-5295},\n}\n\\bib{GrunewaldSegal1}{article}{\n author={Grunewald, Fritz},\n author={Segal, Daniel},\n title={Some general algorithms. I. Arithmetic groups},\n journal={Ann. of Math. (2)},\n volume={112},\n date={1980},\n number={3},\n pages={531--583},\n issn={0003-486X},\n}\n\\bib{GrunewaldSegal2}{article}{\n author={Grunewald, Fritz},\n author={Segal, Daniel},\n title={Decision problems concerning $S$-arithmetic groups},\n journal={J. Symbolic Logic},\n volume={50},\n date={1985},\n number={3},\n pages={743--772},\n issn={0022-4812},\n}\n\\bib{hk}{article}{\n author={Heidrich, Holger},\n author={Kn{\\\"o}ller, Friedrich W.},\n title={\\\"Uber die Fundamentalgruppen Siegelscher Modulvariet\\\"aten vom\n Grade $2$},\n journal={Manuscripta Math.},\n volume={57},\n date={1987},\n number={3},\n pages={249--262},\n issn={0025-2611},\n}\n\\bib{hs}{article}{\n author={Hulek, K.},\n author={Sankaran, G. 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S.},\n title={The congruence subgroup problem},\n conference={\n title={Algebra, $K$-theory, groups, and education},\n address={New York},\n date={1997},\n },\n book={\n series={Contemp. Math.},\n volume={243},\n publisher={Amer. Math. Soc.},\n place={Providence, RI},\n },\n date={1999},\n pages={175--188},\n}\n\\bib{RemyThuillierWernerI}{article}{\n author={R{\\'e}my, Bertrand},\n author={Thuillier, Amaury},\n author={Werner, Annette},\n title={Bruhat-Tits theory from Berkovich's point of view. I. Realizations\n and compactifications of buildings},\n journal={Ann. Sci. \\'Ec. Norm. Sup\\'er. (4)},\n volume={43},\n date={2010},\n number={3},\n pages={461--554},\n issn={0012-9593},\n}\n\\bib{RemyThuillierWernerII}{article}{\n author={R{\\'e}my, Bertrand},\n author={Thuillier, Amaury},\n author={Werner, Annette},\n title={Bruhat-Tits theory from Berkovich's point of view. II. Satake\n compactifications of buildings},\n date={2009},\n eprint={\\tt arXiv:0907.3264 [math.GR]},\n}\n\\bib{san}{article}{\n author={Sankaran, G. K.},\n title={Fundamental group of locally symmetric varieties},\n journal={Manuscripta Math.},\n volume={90},\n date={1996},\n number={1},\n pages={39--48},\n issn={0025-2611},\n}\n\\bib{sap1}{article}{\n author={Saper, Leslie},\n title={Tilings and finite energy retractions of locally symmetric spaces},\n journal={Comment. Math. Helv.},\n volume={72},\n date={1997},\n number={2},\n pages={167--202},\n issn={0010-2571},\n}\n\\bib{sap2}{article}{\n author={Saper, Leslie},\n title={Geometric rationality of equal-rank Satake compactifications},\n journal={Math. Res. Lett.},\n volume={11},\n date={2004},\n number={5},\n pages={653--671},\n issn={1073-2780},\n}\n\\bib{sat1}{article}{\n author={Satake, Ichir{\\^o}},\n title={On representations and compactifications of symmetric Riemannian\n spaces},\n journal={Ann. of Math. (2)},\n volume={71},\n date={1960},\n pages={77--110},\n issn={0003-486X},\n}\n\\bib{sat2}{article}{\n author={Satake, Ichir{\\c{o}}},\n title={On compactifications of the quotient spaces for arithmetically\n defined discontinuous groups},\n journal={Ann. of Math. (2)},\n volume={72},\n date={1960},\n pages={555--580},\n issn={0003-486X},\n}\n\\bib{SerreBourbaki}{article}{\n author={Serre, Jean-Pierre},\n title={Groupes de congruence (d'apr\\`es H. Bass, H. Matsumoto, J.\n Mennicke, J. Milnor, C. Moore)},\n part={Expos\\'e 330},\n book={\n title={S\\'eminaire Bourbaki},\n subtitle={Volume 1966\/1967, Expos\\'e 313--330},\n publisher={W. A. Benjamin},\n address={New York},\n date={1968},\n },\n reprint={\n title={S\\'eminaire Bourbaki, Vol.\\ 10},\n publisher={Soc. Math. France},\n place={Paris},\n date={1995},\n note={pp. 275--291},\n },\n \n}\n\\bib{se3}{article}{\n author={Serre, Jean-Pierre},\n title={Le probl\\`eme des groupes de congruence pour $\\mathbf{SL}_2$},\n journal={Ann. of Math. (2)},\n volume={92},\n date={1970},\n pages={489--527},\n issn={0003-486X},\n}\n\\bib{SerreTrees}{book}{\n author={Serre, Jean-Pierre},\n title={Trees},\n series={Springer Monographs in Mathematics},\n note={Translated from the French original by John Stillwell;\n Corrected 2nd printing of the 1980 English translation},\n publisher={Springer-Verlag},\n place={Berlin},\n date={2003},\n pages={x+142},\n isbn={3-540-44237-5},\n}\n\\bib{tavgen}{article}{\n author={Tavgen{\\cprime}, O. I.},\n title={Bounded generability of Chevalley groups over rings of $S$-integer\n algebraic numbers},\n journal={Izv. Akad. Nauk SSSR Ser. Mat.},\n volume={54},\n date={1990},\n number={1},\n pages={97--122, 221--222},\n issn={0373-2436},\n translation={\n journal={Math. USSR, Izv.},\n volume={36},\n date={1991},\n number={1},\n pages={101--128},\n issn={0025-5726},\n },\n}\n\\bib{Tits}{article}{\n author={Tits, J.},\n title={Reductive groups over local fields},\n conference={\n title={Automorphic forms, representations and $L$-functions (Proc.\n Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part\n 1},\n },\n book={\n series={Proc. Sympos. Pure Math., XXXIII},\n publisher={Amer. Math. Soc.},\n place={Providence, R.I.},\n },\n date={1979},\n pages={29--69},\n}\n\\bib{Venkataramana}{article}{\n author={Venkataramana, T. N.},\n title={On systems of generators of arithmetic subgroups of higher rank\n groups},\n journal={Pacific J. Math.},\n volume={166},\n date={1994},\n number={1},\n pages={193--212},\n issn={0030-8730},\n}\n\\bib{Werner}{article}{\n author={Werner, Annette},\n title={Compactifications of Bruhat-Tits buildings associated to linear\n representations},\n journal={Proc. Lond. Math. Soc. (3)},\n volume={95},\n date={2007},\n number={2},\n pages={497--518},\n issn={0024-6115},\n doi={10.1112\/plms\/pdm019},\n}\n\\bib{wo}{article}{\n author={Wohlfahrt, Klaus},\n title={An extension of F. Klein's level concept},\n journal={Illinois J. Math.},\n volume={8},\n date={1964},\n pages={529--535},\n issn={0019-2082},\n}\n\\bib{Zu1}{article}{\n author={Zucker, Steven},\n title={$L\\sb{2}$ cohomology of warped products and arithmetic groups},\n journal={Invent. Math.},\n volume={70},\n date={1982},\n number={2},\n pages={169--218},\n issn={0020-9910},\n}\n\\bib{Zu2}{article}{\n author={Zucker, Steven},\n title={Satake compactifications},\n journal={Comment. Math. Helv.},\n volume={58},\n date={1983},\n number={2},\n pages={312--343},\n issn={0010-2571},\n}\n\\bib{Zu3}{article}{\n author={Zucker, Steven},\n title={$L\\sb 2$-cohomology and intersection homology of locally symmetric\n varieties, II},\n journal={Compositio Math.},\n volume={59},\n date={1986},\n number={3},\n pages={339--398},\n issn={0010-437X},\n}\n\\end{biblist}\n\\end{bibdiv}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Derivation of cooperative manager gradients} \\label{sec:derivation}\n\nIn this section, we derive an analytic expression of the gradient of the manager policy in a two-level goal-conditioned hierarchy with respect to both the losses associated with the high level and low level policies. In mathematical terms, we are trying to derive an expression for the weighted summation of the derivation of both losses, expressed as follows:\n\\begin{equation}\n \\nabla_{\\theta_m} J_m' = \\nabla_{\\theta_m} \\left( J_m + \\lambda J_w \\right) = \\nabla_{\\theta_m} J_m + \\lambda \\nabla_{\\theta_m} J_w\n \n\\end{equation}\nwhere $\\lambda$ is a weighting term and $J_m$ and $J_w$ are the expected returns assigned to the manager and worker policies, respectively. More specifically, these two terms are:\n\\begin{equation}\n \\resizebox{!}{12pt}{$\n J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^{T\/k} \\gamma^t r_m(s_{kt}) \\right] = \\int_{\\mathcal{S}} \\rho_0(s_t) V_m(s_t) ds_t\n $}\n\\end{equation}\\\\[-25pt]\n\\begin{equation}\n \\resizebox{!}{12pt}{$\n J_w = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\sum_{t=0}^k \\gamma^t r_w(s_t, g_t,\\pi_w(s_t,g_t)) \\right] = \\int_{\\mathcal{S}} \\rho_0(s_t) V_w(s_t, g_t) ds_t \n $}\n\\end{equation}\nHere, under the actor-critic formulation we replace the expected return under a given starting state with the value functions $V_m$ and $V_w$ This is integrated over the distribution of initial states $\\rho_0(\\cdot)$.\n\nFollowing the results by \\cite{silver2014deterministic}, we can express the first term in Eq.~\\eqref{eq:connected-gradient} as:\n\\begin{equation}\n \\nabla_{\\theta_m} J_m = \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)}\\nabla_{\\theta_m} \\pi_m(s) \\right]\n\\end{equation}\n\nWe now expand the second term of the gradient into a function of the manager and worker actor ($\\pi_m$, $\\pi_w$) and critic ($Q_m$, $Q_w$) policies and their trainable parameters.\nIn order to propagate the loss associated with the worker through the policy parameters of the manager, we assume that the goals assigned to the worker $g_t$ are not fixed variables, but rather temporally abstracted outputs from the manager policy $\\pi_m$, and may be updated in between decisions by the manager via a transition function $h$. Mathematically, the goal transition is defined as: \n\\begin{equation}\n g_t(\\theta_m) = \n \\begin{cases}\n \\pi_m(s_t) & \\text{if } t \\text{ mod } k = 0 \\\\\n h(s_{t-1}, g_{t-1}(\\theta_m), s_t) & \\text{otherwise}\n \\end{cases}\n\\end{equation}\nFor the purposes of simplicity, we express the manager output term as $g_t$ from now on.\n\nWe begin by computing the partial derivative of the worker value function with respect to the parameters of the manager:\n\\begin{equation}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} V_w(s_t, g_t) &=\\nabla_{\\theta_m} Q_w (s_t, g_t, \\pi_w(s_t, g_t)) \\\\\n &= \\nabla_{\\theta_m} \\bigg( r_w(s_t, g_t, \\pi_w(s_t,g_t)) +\\int_{\\mathcal{G}} \\int_{\\mathcal{S}} \\gamma p_w(s', g'| s_t,g_t, \\pi_w(s_t,g_t)) V_w(s',g')ds'dg' \\bigg) \\\\\n &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}}\\int_{\\mathcal{S}} p_w(s',g'| s_t, g_t, \\pi_w(s_t,g_t)) V_w(s',g')ds'dg'\n \\end{aligned}\n $}\n \\label{eq:gradient_p1}\n\\end{equation}\nwhere $\\mathcal{G}$ and $\\mathcal{S}$ are the goal and environment state spaces, respectively, and $p_w(\\cdot, \\cdot | \\cdot, \\cdot, \\cdot)$ is the probability distribution of the next state from the perspective of the worker given the current state and action.\n\nExpanding the latter term, we get:\n\\begin{equation}\n \\begin{aligned}\n &p_w(s',g'|s_t,g_t,\\pi_w(s_t,g_t)) = p_{w,1} (g'| s', s_t,g_t,\\pi_w(s_t,g_t)) p_{w,2} (s'| s_t,g_t,\\pi_w(s_t,g_t))\n \\end{aligned}\n \\label{eq:pw_decompose}\n\\end{equation}\nThe first element, $p_{w1}$, is the probability distribution of the next goal, and is deterministic with respect to the conditional variables. Specifically:\n\\begin{equation}\n p_{w,1} (g'| s_t,g_t,\\pi_w(s_t,g_t)) = \n \\begin{cases}\n 1 & \\text{if } g' = g_{t+1} \\\\\n 0 & \\text{otherwise}\n \\end{cases}\n \\label{eq:pw1}\n\\end{equation}\n\nThe second element, $p_{w,2}$, is the state transition probability from the MDP formulation of the task, i.e.\n\\begin{equation}\n p_{w,2}(s'| s_t,g_t,\\pi_w(s_t,g_t)) = p (s'| s_t,\\pi_w(s_t,g_t))\n \\label{eq:pw2}\n\\end{equation}\n\nCombining Eq.~\\eqref{eq:pw_decompose}-\\eqref{eq:pw2} into Eq.~\\eqref{eq:gradient_p1}, we get:\n\\begin{equation} \\label{eq:simplified-next-step-value}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} V_w(s_t,g_t) &=\\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) \\\\\n &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}}\\int_{\\mathcal{S}}\\bigg( p_{w,1} (g'| s', s_t,g_t,\\pi_w(s_t,g_t)) p_{w,2} (s'| s_t,g_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg'\\bigg) \\\\\n %\n &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) \\\\\n &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{G}\\cap \\{g_{t+1}\\}}\\int_{\\mathcal{S}} 1 \\cdot p (s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg' \\\\\n &\\quad + \\gamma \\nabla_{\\theta_m} \\int_{(\\mathcal{G}\\cap \\{g_{t+1}\\})^c}\\int_{\\mathcal{S}} 0 \\cdot p (s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g') ds'dg' \\\\\n %\n &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) + \\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{S}} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(s',g_{t+1})ds'\n \\end{aligned}\n $}\n\\end{equation}\n\nContinuing the derivation of $\\nabla_{\\theta_m}V_w$ from Eq.~\\eqref{eq:simplified-next-step-value}, we get,\n\\begin{equation} \\label{eq:continue-derivatione}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} V_w(s_t,g_t) &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) +\\gamma \\nabla_{\\theta_m} \\int_{\\mathcal{S}} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(g_{t+1}, s')ds' \\\\\n %\n &= \\nabla_{\\theta_m} r_w(s_t,g_t,\\pi_w(s_t,g_t)) +\\gamma \\int_{\\mathcal{S}} \\nabla_{\\theta_m} p(s'| s_t,\\pi_w(s_t,g_t)) V_w(g_{t+1}, s')ds' \\\\\n %\n &= \\nabla_{\\theta_m} g_t \\nabla_g r_w(s_t,g,\\pi_w(s_t,g_t))|_{g=g_t} \\\\\n &\\quad + \\nabla_{\\theta_m}g_t \\nabla_g \\pi_w (s_t,g)|_{g=g_t} \\nabla_a r_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)} \\\\\n &\\quad +\\gamma\\int_\\mathcal{S} \\bigg(V_w(s',g_{t+1})\\nabla_{\\theta_m} g_t \\nabla_g \\pi_w(s_t,g)|_{g=g_t} \\nabla_a p(s'\\vert s_t,a)|_{a=\\pi_w(s_t,g_t)}ds'\\bigg)\\\\\n &\\quad +\\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n %\n &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) \\\\\n &\\quad \\quad \\quad \\quad \\quad \\quad + \\pi_w (s_t,g) \\nabla_a r_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)} \\vphantom{\\int} \\\\\n &\\quad \\quad \\quad \\quad \\quad \\quad + \\gamma\\int_\\mathcal{S} V_w(s',g_{t+1}) \\pi_w(s_t,g) \\nabla_a p(s'\\vert s_t,a)|_{a=\\pi_w(s_t,g_t)}ds' \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n &\\quad +\\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n %\n &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) \\\\\n &\\quad \\quad \\quad \\quad \\quad \\quad + \\pi_w (s_t,g) \\nabla_a \\bigg( r_w(s_t,g_t,a) + \\gamma\\int_\\mathcal{S} V_w(s',g_{t+1}) p(s'\\vert s_t,a)ds' \\bigg)\\bigg\\rvert_{a=\\pi_w(s_t,g_t)} \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n &\\quad + \\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(g_t, s_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\\\\\n &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\n \\\\\n &\\quad + \\gamma\\int_\\mathcal{S}p(s'\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s',g_{t+1}) ds'\n \\end{aligned}\n $}\n\\end{equation} \n\nIterating this formula, we have,\n\\begin{equation}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} V_w(s_t,g_t) &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\\\\\n &\\quad +\\gamma\\int_\\mathcal{S}p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t))\\nabla_{\\theta_m} V_w(s_{t+1},g_{t+1}) ds_{t+1} \\\\\n %\n &= \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t}\n \\quad \\\\\n &\\quad +\\gamma\\int_\\mathcal{S}p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t)) \\nabla_{\\theta_m} g_{t+1} \\nabla_g \\bigg(r_w(s_{t+1},g,\\pi_w(s_{t+1},g_{t+1})) \\vphantom{\\int} \\\\\n &\\quad \\quad \\quad \\quad \\quad + \\pi_w (s_{t+1},g) \\nabla_a Q_w(s_{t+1},g_{t+1},a)|_{a=\\pi_w(s_{t+1},g_{t+1})}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_{t+1}}ds_{t+1} \\\\\n & \\quad +\\gamma^2 \\int_\\mathcal{S}\\int_\\mathcal{S} \\bigg( p(s_{t+1}\\vert s_t,\\pi_w(s_t,g_t)) p(s_{t+2}\\vert s_{t+1},\\pi_w(g_{t+1}, s_{t+1}))\\\\\n &\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\nabla_{\\theta_m} V_w(s_{t+2},g_{t+2}) ds_{t+2} ds_{t+1} \\bigg)\\\\\n \n & \\hspace{45mm} \\vdots\\\\\n &= \\sum_{n=0}^{\\infty} \\gamma^n \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n \\text{ times}} \\left(\\prod_{k=0}^{n-1} p(s_{t+k+1}|s_{t+k},\\pi_w(s_{t+k},g_{t+k})) \\right) \\\\\n &\\quad \\quad \\quad \\quad \\times \\nabla_{\\theta_m} g_{t+n} \n \\nabla_g \\bigg(r_w(s_{t+n},g,\\pi_w(s_{t+n},g_{t+n})) \\\\\n & \\quad \\quad \\quad +\\pi_w (s_{t+n},g) \\nabla_a Q_w(s_{t+n},g_{t+n},a)|_{a=\\pi_w(s_{t+n},g_{t+n})}\\bigg)\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_{t+n}} ds_{t+n}\\cdots ds_{t+1}\n \\end{aligned}\n $}\n\\end{equation}\n\nTaking the gradient of the expected worker value function, we get,\n\\begin{small}\n\\begin{equation}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} J_w &= \\nabla_{\\theta_m} \\int_{\\mathcal{S}} \\rho_0(s_0) V_w(s_0, g_0) ds_0 \\\\\n %\n &= \\int_{\\mathcal{S}} \\rho_0(s_0) \\nabla_{\\theta_m} V_w(s_0, g_0) ds_0 \\\\\n %\n &= \\int_{\\mathcal{S}} \\rho_0(s_0) \\sum_{n=0}^{\\infty} \\gamma^n \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n \\text{ times}} \\Bigg[\\left(\\prod_{k=0}^{n-1} p(s_{k+1}|s_k,\\pi_w(s_k,g_k)) \\right) \\nabla_{\\theta_m} g_n \\\\\n &\\quad \\quad \\quad \\quad \\times \\nabla_g \\bigg(r_w(s_n,g,\\pi_w(s_n,g_n))\\vphantom{\\int} + \\pi_w (s_n,g) \\nabla_a Q_w(s_n,g_n,a)|_{a=\\pi_w(s_n,g_n)}\\vphantom{\\int} \\bigg)\\Bigg] \\bigg\\rvert_{g=g_n} ds_n\\cdots ds_0 \\\\\n %\n &= \\sum_{n=0}^{\\infty} \\underbrace{\\int_\\mathcal{S} \\cdots \\int_\\mathcal{S}}_{n+1 \\text{ times}} \\gamma^n p_{\\theta_m, \\theta_w, n}(\\tau) \\nabla_{\\theta_m} g_n\n \\nabla_g \\bigg(r_w(s_n,g,\\pi_w(s_n,g_n))\\vphantom{\\int}\\\\\n &\\quad \\quad \\quad \\quad + \\pi_w (s_n,g) \\nabla_a Q_w(s_n,g_n,a)|_{a=\\pi_w(s_n,g_n)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_n} ds_n\\cdots ds_0 \\\\\n \n &= \\mathbb{E}_{\\tau \\sim p_{\\theta_m, \\theta_w}(\\tau)} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t} \\bigg]\n \\end{aligned}\n $}\n\\end{equation}\n\\end{small}\nwhere $\\tau=(s_0, a_0, s_1, a_1, \\dots, s_n)$ is a trajectory and $p_{\\theta_m, \\theta_w, n}(\\tau)$ is the (improper) discounted probability of witnessing a trajectory a set of policy parameters $\\theta_m$ and $\\theta_w$.\n\nThe final representation of the connected gradient formulation is then:\n\\begin{equation}\n \\resizebox{.9\\hsize}{!}{$\n \\begin{aligned}\n \\nabla_{\\theta_m} J_m' &= \\mathbb{E}_{s\\sim p_\\pi} \\left[ \\nabla_a Q_m (s,a)|_{a=\\pi_m(s)}\\nabla_{\\theta_m} \\pi_m(s) \\right] \\\\\n & \\quad + \\mathbb{E}_{\\tau \\sim p_{\\theta_m, \\theta_w}(\\tau)} \\bigg[ \\nabla_{\\theta_m} g_t \\nabla_g \\bigg(r_w(s_t,g,\\pi_w(s_t,g_t)) + \\pi_w (s_t,g) \\nabla_a Q_w(s_t,g_t,a)|_{a=\\pi_w(s_t,g_t)}\\vphantom{\\int} \\bigg) \\bigg\\rvert_{g=g_t} \\bigg]\n \\end{aligned}\n $}\n\\end{equation}\n\n\n\\section{Cooperative HRL as goal-constrained optimization}\n\\label{sec:constrained-hrl}\n\nIn this section we will derive a constrained optimization problem that motivates cooperation between a meta policy $\\pi$ and a worker policy $\\omega$. We will derive an update rule for the finite horizon reinforcement learning setting, and then approximate the derivation for stationary policies by dropping the time dependencies from the meta policy, worker policy, and the cooperative $\\lambda$. Our goal is to find a hierarchy of policies $\\pi$ and $\\omega$ with maximal expected return subject to a constraint on minimum expected distance from goals proposed by $\\pi$. Put formally, \n\\begin{gather}\n \\max_{\\pi_{0:T}, \\omega_{0:T}} \\sum_{t = 0}^{T} \\mathbb{E} \\left[ r (s_{t}, a_{t}) \\right] \\;\\text{s.t.}\\; \\sum_{i = t}^{T} \\mathbb{E} \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right] \\leq \\delta \\; \\forall t\n\\end{gather}\n\nwhere $\\delta$ is the desired minimum expected distance from goals proposed by $\\pi$. The optimal worker policy $\\omega$ without the constraint need not be goal-reaching, and so we expect the constraint to be tight in practice---this seems to be true in our experiments in this paper. The hierarchy of policies at iteration $t$ may only affect the future, and so we can use approximate dynamic programming to solve for the optimal hierarchy at the last timestep, and proceed backwards in time. We write the optimization problem as iterated maximization,\n\\begin{gather}\n \\max_{\\pi_{0}, \\omega_{0}} \\mathbb{E} \\left[ r (s_{0}, a_{0}) + \\max_{\\pi_{1}, \\omega_{1}} \\mathbb{E} \\left[ \\cdots + \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right] \\right] \\right]\n\\end{gather}\n\nsubject to a constraint on the minimum expected distance from goals proposed by $\\pi$. Starting from the last time step, we convert the primal problem into a dual problem. Subject to the original constraint on minimum expected distance from goals proposed by $\\pi_{T}$ at the last timestep,\n\\begin{gather}\n \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right] = \\min_{\\lambda_{T} \\geq 0} \\max_{\\pi_{T}, \\omega_{T}} \\mathbb{E} \\left[ r (s_{T}, a_{T}) \\right] + \\lambda_{T} \\delta - \\lambda_{T} \\sum_{i = T}^{T} \\mathbb{E} \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right]\n\\end{gather}\n\nwhere $\\lambda_{T}$ is a Lagrange multiplier for time step $T$, representing the extent of the cooperation bonus between the meta policy $\\pi_{T}$ and the worker policy $\\omega_{T}$ at the last time step. In the last step we applied strong duality, because the objective and constraint are linear functions of $\\pi_{T}$ and $\\omega_{T}$. Solving the dual problem corresponds to CHER, which trains a meta policy $\\pi_{T}$ with a cooperative goal-reaching bonus weighted by $\\lambda_{T}$. The optimal cooperative bonus can be found by performing minimization over a simplified objective using the optimal meta and worker policies.\n\\begin{gather}\n \\min_{\\lambda_{T}\\geq 0} \\lambda_{T} \\delta - \\lambda_{T} \\sum_{i = T}^{T} \\mathbb{E}_{g_{i} \\sim \\pi^{*}_{T} (g_{i} | s_{i}; \\lambda_{T}), a_{i} \\sim \\omega^{*}_{T} (a_{i} | s_{i}, g_{i}; \\lambda_{T}) } \\left[ \\left\\| s_{i + 1} - g_{i} \\right\\|_{p} \\right]\n\\end{gather}\n\nBy recognizing that in the finite horizon setting the expected sum of rewards is equal to the meta policy's Q function and the expected sum of distances to goals is the worker policy's Q function for deterministic policies, we can separate the dual problem into a bi-level optimization problem first over the policies. \n\\begin{gather}\n \\max_{\\pi_{T}, \\omega_{T}} Q_{m}(s_{T}, g_{T}, a_{T}) - \\lambda_{T} Q_{w}(s_{T}, g_{T}, a_{T})\\\\\n \\min_{\\lambda_{T}\\geq 0} \\lambda_{T} \\delta + \\lambda_{T} Q_{w}(s_{T}, g_{T}, a_{T})\n\\end{gather}\n\nBy solving the iterated maximization backwards in time, solutions for $t