metriques de carnot cartheodory et quasiisometries des espaces symetriques de rang un we exhibit a rigidity property of the simple groups sp n 1 and f7 20 which implies mostow rigidity this property does not extend to o n 1 and u n 1 the proof relies on quasiconformal theory applied in the cr setting extensions are given to a class of solvable lie groups as a byproduct a result on quasiisometries of infinite nilpotent groups is obtained dans cet article on etablit une propriete de rigidite des groupes simples de rang un sp n 1 n 2 2 et fqui entraine la rigidite de mostow 1 theoreme toute quasiisornwtrie de 1 espace hyperbolique quaternionien hhn n 2 resp du plan hyperbolique de cayley cah2 est d distance bornee d une isometrie i e differe d une isomnetrie par une application qui diplace les points d une distance bornee une application f entre espaces metriques est une quasiisometrie s il existe des constantes l et c telles que l image de f soit c dense et que pour tous x y c d x y d fx fy ld x y c l une quasiisometrie entre des g et g est une sorte d isomorphisme virtuel dans la categorie topologique en effet cela correspond a une action de g sur un fibre principal c de groupe g sur une base compacte cf ra un isomorphisme entre sous groupes cocompacts covolume fini suffit pour les groupes simples de rang un de groupes de lie s etend en une quasiisometrie des espaces symetriques ou des groupes de lie si celle ci est proche d une isometrie des espaces symetriques resp un isomorphisme des groupes de lie les sous groupes sont conjugues c est la rigidite de mostow m2 la propriete ci dessus ne s etend pas aux groupes o n 1 et u n 1 en effet paragraphe 11 7 ceux ci ont beaucoup de quasiisometries au moins this content downloaded from 207 46 13 92 on sun 20 nov 2016 04 26 46 utc all use subject to http about jstor org terms

gl equivariant modules over polynomial rings in infinitely many variables consider the polynomial ring in countably infinitely many variables over a field of characteristic zero together with its natural action of the infinite general linear group g we study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible g action we define and prove finiteness properties for analogues of hilbert series systems of parameters depth local cohomology koszul duality and regularity we also show that this category is built out of a simpler more combinatorial quiver category which we describe explicitly our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions for example the paper by church ellen berg and farb on the category of fi modules which is equivalent to our category along the way we see several connections with the character polynomials from the representation theory of the symmetric groups several examples are given to illustrate that the invariants we introduce are explicit and computable

compact subgroups of linear algebraic groups abstract the general problem underlying this article is to give a qualitative classification of all compact subgroups gl n f where f is a local field and n is arbitrary it is natural to ask whether is an open compact subgroup of h e where h is a linear algebraic group over a closed subfield e f we show that indeed has this form up to finite index and a finite number of abelian subquotients when is zariski dense in a connected semisimple group we give a precise openness result for the closure of the commutator group of in the case char f 0 the answers have long been known by results of chevalley and weyl the motivation for this work comes from the positive characteristic case where such results are needed to study galois representations associated to function fields we also derive openness results over a finite number of local fields

density of the commensurability groups of uniform tree lattices abstract let x be a universal cover of a finite connected graph and a group acting discretely and cocompactly on x i e a uniform lattice on x we announce a proof of the conjecture of h bass and r kulkami j amer math soc no 4 1990 843 902 that the commensurability group of is dense in the automorphism group of x

enumeration of points lines planes etc one of the earliest results in enumerative combinatorial geometry is the following theorem of de bruijn and erd s every set of points e in a projective plane determines at least lvert e rvert lines unless all the points are contained in a line the result was extended to higher dimensions by motzkin and others who showed that every set of points e in a projective space determines at least lvert e rvert hyperplanes unless all the points are contained in a hyperplane let e be a spanning subset of an r dimensional vector space we show that in the partially ordered set of subspaces spanned by subsets of e there are at least as many r p dimensional subspaces as there are p dimensional subspaces for every p at most frac 1 2 r this confirms the top heavy conjecture by dowling and wilson for all matroids realizable over some field the proof relies on the decomposition theorem package for ell adic intersection complexes

generalized ccr flows we introduce a new construction of e0 semigroups called generalized ccr flows with two kinds of descriptions those arising from sum systems and those arising from pairs of c0 semigroups we get a new necessary and sufficient condition for them to be of type iii when the associated sum system is of finite index using this criterion we construct examples of type iii e0 semigroups which can not be distinguished from e0 semigroups of type i by the invariants introduced by boris tsirelson finally by considering the local von neumann algebras and by associating a type iii factor to a given type iii e0 semigroup we show that there exist uncountably many type iii e0 semigroups in this family which are mutually non cocycle conjugate

van kampen s embedding obstruction for discrete groups we give a lower bound to the dimension of a contractible mani fold on which a given group can act properly discontinuously in particular we show that the n fold product of nonabelian free groups cannot act prop erly discontinuously on r 2n 1

topological equivalence of flows on homogeneous spaces and divergence of one parameter subgroups of lie groups soient et des treillis et et des sous groupes a un parametre des groupes de lie connexes g et g si une des conditions a b ou c est vraie alors le theoreme a etablit que si les flots induits sur les espaces homogenes g et g sont topologiquement equivalents alors ils sont topologiquement equivalents par une application affine

planar groups and the seifert conjecture we describe a number of characterisations of virtual surface groups which are based on the following result let g be a group and f be a field we show that if g is fp2 over f and if h2 g f thought of as a k vector space contains a 1 dimensional ginvariant subspace then g is a virtual surface group i e contains a subgroup of finite index which is the fundamental group of a closed surface other than the sphere or projective plane in particular this applies to rational poincare duality groups we also conclude that a finitely presented group which is semistable at infinity and with infinite cyclic fundamental group at infinity is a virtual surface group we recover the result of mess which characterises such groups as groups which are quasiisometric to complete riemannian planes we also give a cohomological version of the seifert conjecture from which the topological seifert conjecture proven by tukia mess gabai casson and jungreis can be recovered via work of zieschang and scott

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

some examples of quasiisometries of nilpotent lie groups we construct quasiisometries of nilpotent lie groups in particular for any simply connected nilpotent lie group n we construct quasiisometries from n to itself that is not at finite distance from any map that is a composition of left translations and automorphisms we also construct bilipschitz maps of the heisenberg groups that send vertical lines to curves that are not vertical lines

the growth of linear groups abstract we present a short self contained relatively simple proof to the growth dichotomy of linear groups the proof depends on the fixed point property of amenable groups and the main tool is furstenberg s lemma regarding measures on projective spaces

minimal time minimal order deadbeat regulator with internal stability this paper is concerned with the output deadbeat finite settling time regulation with internal stability for linear discrete time multivariable systems the two cases are treated separately the one is the regulation by state feedback and the other is the regulation through function observers for both cases the basic solvability conditions the minimal settling time and the characterizations of the minimal time regulators are derived throughout the paper the well posedness condition plays a fundamental role a deterministic version of the so called separation theorem in lqg problem is derived namely the minimal time state feedback regulator combined with the minimal time minimal order function observer yields the overall minimal time minimal order regulator by output feedback

self tuning deadbeat controllers abstract a deterministic self tuning deadbeat control scheme is presented which is extremely simple to implement in the single variable and multivariable cases its convergence properties are analysed by straightforward application of de larminat s convergence results

topological equivalence of foliations of homogeneous spaces for i 1 2 let fi be a lattice in a connected lie group gi and let xi be a connected lie subgroup of gi the double cosets rigxi provide a foliation jt of the homogeneous space fi gi assume that xi and x2 are unimodular and that jr has a dense leaf if gi and g2 are semisimple groups to which the mostow rigidity theorem applies or are simply connected nilpotent groups or are certain more general solvable groups we use an idea of d benardete to show that any topological equivalence of jr and 7 must be the composition of two very elementary maps an affine map and a map that takes each leaf to itself

relative property t and related properties of wreath products let h and be two discrete groups and x be a set on which acts we investigate when the wreath product h x has relative property t and give a complete characterization of property t for such groups moreover we determine a lower bound of a kazhdan constant for h x in terms of those for h and property of h x is also discussed in some instances

superrigidity of actions on finite rank median spaces abstract finite rank median spaces are a simultaneous generalisation of finite dimensional cat 0 cube complexes and real trees if is an irreducible lattice in a product of rank one simple lie groups we show that every action of on a complete finite rank median space has a global fixed point this is in sharp contrast with the behaviour of actions on infinite rank median spaces where even proper cocompact actions can arise the fixed point property is obtained as corollary to a superrigidity result the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups we exploit roller compactifications of median spaces these were introduced in 48 and generalise a well known construction in the case of cube complexes we show that the haagerup cocycle provides a reduced 1 cohomology class that detects group actions with a finite orbit in the roller compactification this is new even for cat 0 cube complexes and has interesting consequences involving shalom s property h f d for instance in gromov s density model random groups at low density do not have h f d

on the quantitative quasi isometry problem transport of poincare inequalities and different types of quasi isometric distortion growth abstract we consider a quantitative form of the quasi isometry problem we discuss several arguments which lead us to a number of results and bounds of quasi isometric distortion comparison of volumes connectivity etc then we study the transport of poincare constants by quasi isometries and we give sharp lower and upper bounds for the homotopy distortion growth for a certain class of hyperbolic metric spaces a quotient of a heintze group r r n by z n we also prove the linear distortion growth between hyperbolic space h n n 3 and a tree

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

on deadbeat controllers the possibility of achieving a deadbeat regulation and tracking in linear multivariable systems independently of their initial conditions is investigated necessary and sufficient conditions for the existence of various deadbeat controllers are established in a constructive way a detailed analysis of the control sequence generated by any deadbeat controller is then presented

quasi flats and rigidity in higher rank symmetric spaces in this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces such as sl n 2 n 3 theorem 8 1 is a major step towards the proof of quasiisometric rigidity of such lattices e we also give a different and effective proof of the theorem of kleiner leeb on the quasi isometric rigidity of higher rank symmetric spaces kl

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

almost invariance of distributions for random walks on groups we study the neighborhoods of a typical point z n visited at n th step of a random walk determined by the condition that the transition probabilities stay close to mu n z n if such neighborhood contains a ball of radius c sqrt n we say that the random walk has almost invariant transition probabilities we prove that simple random walks on wreath products of mathbb z with finite groups have almost invariant distributions a weaker version of almost invariance implies a necessary condition of ozawa s criterion for the property h mathrm fd we define and study the radius of almost invariance we estimate this radius for random walks on iterated wreath products and show this radius can be asymptotically strictly smaller than n l n where l n denotes the drift function of the random walk we show that the radius of individual almost invariance of a simple random walk on the wreath product of mathbb z 2 with a finite group is asymptotically strictly larger than n l n finally we show the existence of groups such that the radius of almost invariance is smaller than a given function but remains unbounded we also discuss possible limiting distribution of ratios of transition probabilities on non almost invariant scales

orlicz spaces and the large scale geometry of heintze groups we consider an orlicz space based cohomology for metric measured spaces with bounded geometry we prove the quasi isometry invariance for a general young function in the hyperbolic case we prove that the degree one cohomology can be identified with an orlicz besov function space on the boundary at infinity we give some applications to the large scale geometry of homogeneous spaces with negative curvature heintze groups as our main result we prove that if the heintze group is not of carnot type any self quasi isometry fixes a distinguished point on the boundary and preserves a certain foliation on the complement of that point

quasi actions on trees i bounded valence given a bounded valence bushy tree t we prove that any cobounded quasi action of a group g on t is quasiconjugate to an action of g on another bounded valence bushy tree t this theorem has many applications quasi isometric rigidity for fundamental groups of finite bushy graphs of coarse pd n groups for each fixed n a generalization to actions on cantor sets of sullivan s theorem about uniformly quasiconformal actions on the 2 sphere and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice finally we give the first examples of two finitely generated groups which are quasi isometric and yet which cannot act on the same proper geodesic metric space properly discontinuously and cocompactly by isometries

group splittings and asymptotic topology it is a consequence of the theorem of stallings on groups with many ends that splittings over finite groups are preserved by quasi isometries in this paper we use asymptotic topology to show that group splittings are preserved by quasi isometries in many cases roughly speaking we show that splittings are preserved under quasi isometries when the vertex groups are fundamental groups of aspherical manifolds or more generally coarse pd n groups and the edge groups are smaller than the vertex groups

a proof of the hilbert smith conjecture for actions by lipschitz maps the classical hilbert 5th problem 14 asks whether every nite dimensional locally euclidean topological group is necessarily a lie group it was solved in the a rmative by von neumann 23 for compact groups in 1933 and by gleason 11 and by montgomery and zippin 20 for locally compact groups in 1952 a more general version of the hilbert 5th problem called the hilbert smith conjecture asserts that among all locally compact groups only lie groups g can act e ectively on nite dimensional manifolds m i e each g g e moves at least one point of m 28 it follows from the work of newman 24 and smith 29 that this conjecture is equivalent to its special case when the acting group g is the group of p adic integers ap in 1946 bochner and montgomery 3 proved the hilbert smith conjecture for groups g acting e ectively on a manifold m by di eomorphisms a simpler geometrical proof was obtained by skopenkov and the authors 25 using the idea of smooth homogeneity a compact subset k m of a smooth manifold m is said to be smoothly ambiently homogeneous i e for each x y k there exists a di eomorphism h m k x m k y it was shown that this property implies that k is a smooth submanifold of m therefore g k is a lie group the proof reveals a close relationship between homogeneity and taming theory for compact subsets of rn which are pinched by tangent balls the latter problem was investigated in the past by various authors 6 10 12 16 17 see also a very interesting paper by hahn 13 an interesting approach to the hilbert smith conjecture is via wild cantor sets in rn with strong homogeneity properties note that the antoine necklace

coarse differentiation of quasi isometries i spaces not quasi isometric to cayley graphs in this paper we prove that certain spaces are not quasi isometric to cayley graphs of nitely generated groups in particular we answer a question of woess and prove a conjecture of diestel and leader by showing that certain homogeneous graphs are not quasi isometric to a cayley graph of a nitely generated group this paper is the rst in a sequence of papers proving results announced in our 2007 article quasi isometries and rigidity of solvable groups in particular this paper contains many steps in the proofs of quasi isometric rigidity of lattices in sol and of the quasi isometry classication of lamplighter groups the proofs of those results are completed in coarse dierentiation of quasi isometries ii rigidity for lattices in sol and lamplighter groups the method used here is based on the idea of coarse dierentiation introduced in our 2007 article

quasisymmetric maps on the boundary of a negatively curved solvable lie group we describe all the self quasisymmetric maps on the ideal boundary of a particular negatively curved solvable lie group as applications we derive some rigidity properties for quasiisometries of the solvable lie group

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

a sierpinski carpet with the co hopfian property motivated by questions in geometric group theory we define a quasisymmetric co hopfian property for metric spaces and provide an example of a metric sierpinski carpet with this property as an application we obtain a quasi isometrically co hopfian gromov hyperbolic space with a sierpinski carpet boundary at infinity in addition we give a complete description of the quasisymmetry group of the constructed sierpinski carpet this group is uncountable and coincides with the group of bi lipschitz transformations

finite grobner bases in infinite dimensional polynomial rings and applications abstract we introduce the theory of monoidal grobner bases a concept which generalizes the familiar notion in a polynomial ring and allows for a description of grobner bases of ideals that are stable under the action of a monoid the main motivation for developing this theory is to prove finiteness results in commutative algebra and applications a basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry using this machinery we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of hosten and the second author

reduced 1 cohomology and relative property t shalom characterized property t in terms of the vanishing of all reduced first cohomology for compactly generated groups we characterize group pairs having the property that the restriction map on all first reduced cohomology vanishes we show that in a strong sense this is inequivalent to relative property t even for compactly generated group pairs

proper actions of lattices on contractible manifolds every lattice in a connected semi simple lie group g acts properly discontinuously by isometries on the contractible manifold g k k a maximal compact subgroup of g we prove that if acts on a contractible manifold w and if either 1 the action is properly discontinuous or 2 w is equipped with a complete riemannian metric the action is by isometries and with unbounded orbits g is simple with finite center and rank 1 then dimw dimg k

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

deadbeat characteristic of one step ahead control abstract the deadbeat characteristic of one step ahead control is examined in detail it has been clarified that the plant output y k coincides with the reference output y m k for all k n 1 n m where n is the order of the plant and n m is the relative degree of the plant transfer function