on the eisenstein cohomology of arithmetic groups the cohomology h gamma e of an arithmetic subgroup gamma of a connected reductive algebraic group g defined over some algebraic number field f can be interpreted in terms of the automorphic spectrum of gamma with this framework in place there is a sum decomposition of the cohomology into the cuspidal cohomology i e classes represented by cuspidal automorphic forms for g and the so called eisenstein cohomology constructed as the span of appropriate residues or derivatives of eisenstein series attached to cuspidal automorphic forms on the levi components of proper parabolic f subgroups of g the main objective of this paper is to isolate a specific structural part in the eisenstein cohomology this pertains to regular eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered it is shown that the cohomological degree of these classes is bounded from below by the constant q 0 g mathbb r 1 2 dim x g mathbb r rk g mathbb r rk k mathbb r where k mathbb r denotes a maximal compact subgroup of the real lie group g mathbb r where x g mathbb r is the associated symmetric space this investigation has various applications one of these is a vanishing result for the cohomology in the generic case i e where the representation determining the coefficient system e has regular highest weight in degrees below q 0 g mathbb r this is a sharp bound depending only on the underlying real lie group g mathbb r corollary 5 6 proposition 5 8 this result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular eisenstein cohomology classes

some groups of transformations defined by jordan algebras ii groups of type f4 the principal objective of this paper is the study of one of the types of linear groups defined in i 15 the groups of automorphisms of central simple exceptional jordan algebras this can also be considered as the second step in a program of studying exceptional linear groups analogous to the exceptional simple lie groups via non associative algebras the first step in this program has been taken in another paper 13 in which we studied the counterparts of the lie groups of type g 2 as groups of automorphisms of cayley algebras in an analogous fashion the present paper deals with the counterparts of the lie group f 4 as groups of automorphisms of exceptional jordan algebras

on the cuspidal cohomology of arithmetic groups it is the main objective of this paper to show a vanishing result for cuspidal cohomology of arithmetic groups in classical groups g defined over some number field k our approach relies on the fact that cuspidal automorphic forms are nonsingular in the sense of howe this result puts a strong constraint on the archimedean components of irreducible cuspidal automorphic representations of g that can possibly contribute to the cuspidal cohomology combining this with the classification of vogan zuckerman of unitary representations with nonzero cohomology provides a constant r 0 g k only depending on g k below which the cuspidal cohomology vanishes we will give a formula for this constant for each classical group of type i in the classification scheme due to weil we conclude with making this result explicit for some split classical groups over a totally real algebraic number field

totally skew embeddings of manifolds we study a version of whitney s embedding problem in projective geometry what is the smallest dimension of an affine space that can contain an n dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points this problem is related to the generalized vector field problem existence of non singular bilinear maps and the immersion problem for real projective spaces we use these connections and other methods to obtain several specific and general bounds for the desired dimension

adsorption and collapse of self avoiding walks in three dimensions a monte carlo study we consider self avoiding walks on the simple cubic lattice with a unit degree vertex at the origin and confined to the half space in addition the walks interact with the plane z 0 and have a vertex vertex interaction within the walk so that the walk can adsorb at the surface z 0 and also undergo collapse we investigate the interaction between collapse and adsorption using monte carlo methods and explore the form of the phase diagram

on the growth of torsion in the cohomology of arithmetic groups in this paper we consider certain families of arithmetic subgroups of mathrm so 0 p q and mathrm sl 3 mathbb r respectively we study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules we show that for natural sequences of such modules the torsion in the cohomology grows exponentially

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

some groups of transformations defined by jordan algebras i it is well known that the lie algebra of derivations of the exceptional jordan algebra m 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional lie algebra f 4 of killing cartan and that the lie algebra of linear transformations in m 3 8 leaving a certain cubic norm form defined on m 3 8 invariant is the exceptional lie algebra e 6 chevalley schafer 7 freudenthal 11 the program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple jordan algebras the present paper gives the definition of the groups and lie algebras in the general case and gives their determination for special jordan algebras with some restrictions on the base field in a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple jordan algebras and in a third paper we hope to study the generalized groups of types e 6 which arise from these jordan algebras

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

on the non vanishing of the first betti number of hyperbolic three manifolds we show the non vanishing of cohomology groups of sufficiently small congruence lattices in sl 1 d where d is a quaternion division algebra defined over a number field e contained inside a solvable extension of a totally real number field as a corollary we obtain new examples of compact arithmetic hyperbolic three manifolds with non torsion first homology group confirming a conjecture of waldhausen the proof uses the characterisation of the image of solvable base change by the author and the construction of cusp forms with non zero cusp cohomology by labesse and schwermer

bounds for eigenforms on arithmetic hyperbolic 3 manifolds on a family of arithmetic hyperbolic 3 manifolds of square free level we prove an upper bound for the sup norm of hecke maas cusp forms with a power saving over the local geometric bound simultaneously in the laplacian eigenvalue and the volume by a novel combination of diophantine and geometric arguments in a noncommutative setting we obtain bounds as strong as the best corresponding results on arithmetic surfaces

on highly regular embeddings a continuous map mathbb r d rightarrow mathbb r n is k regular if it maps any k pairwise distinct points to k linearly independent vectors our main result on k regular maps is the following lower bound for the existence of such maps between euclidean spaces in which k denotes the number of ones in the dyadic expansion of k

collapse of two dimensional linear polymers we study the collapse transition of a two dimensional very long polymer the model we consider is a lattice model where the chain is represented by a self avoiding walk with nearest neighbor attraction by using the transfer matrix technique we calculate exactly the thermal and geometrical properties of the polymer on strips of finite width we then use finite size scaling to determine the values of the tricritical point exponentsv1 0 55 0 01vu 1 15 0 15 1 v1 1 80 0 05 we compare these results to the other values already proposed in the literature

on the torsion in the cohomology of arithmetic hyperbolic 3 manifolds in this paper we consider the cohomology of a closed arithmetic hyperbolic 3 manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of sl 2 c the cohomology is defined over the integers and is a finite abelian group we show that the order of the 2nd cohomology grows exponentially as the local system grows we also consider the twisted ruelle zeta function of a closed arithmetic hyperbolic 3 manifold and we express the leading coefficient of its laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology

examination of the theta point from exact enumeration of self avoiding walks self avoiding walks on the square lattice with nearest neighbour attractive interactions are investigated as a model for the two dimensional version of a polymer in dilute solution temperature dependences of the exponent nu and the free energy of the chain are estimated from the exact enumeration data for up to 20 steps the value of nu at the theta point disagrees with the mean field theory the end distance distribution function at the theta point is also examined

subgroups of bianchi groups and arithmetic quotients of hyperbolic 3 space let o be the ring of integers in an imaginary quadratic number field the group psl 2 o acts discontinuously on hyperbolic 3 space h if psl 2 o is a torsionfree subgroup of finite index then the manifold h can be compactified to a manifold m so that the inclusion h m is a homotopy equivalence m is a compact with boundary the boundary being a union of finitely many 2 tori this paper contains a computer aided study of subgroups of low index in psl 2 o for various o the explicit description of these subgroups leads to a study of the homeomorphism types of the m

skew flat fibrations a fibration of mathbb r n by oriented copies of mathbb r p is called skew if no two fibers intersect nor contain parallel directions conditions on p and n for the existence of such a fibration were given by ovsienko and tabachnikov a classification of smooth fibrations of mathbb r 3 by skew oriented lines was given by salvai in analogue with the classification of oriented great circle fibrations of s 3 by gluck and warner we show that salvai s classification has a topological variation which generalizes to characterize all continuous fibrations of mathbb r n by skew oriented copies of mathbb r p we show that the space of fibrations of mathbb r 3 by skew oriented lines deformation retracts to the subspace of hopf fibration and therefore has the homotopy type of a pair of disjoint copies of s 2 we discuss skew fibrations in the complex and quaternionic setting and give a necessary condition for the existence of a fibration of mathbb c n respectively mathbb h n by skew oriented copies of mathbb c p respectively mathbb h p

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

surface subgroups of kleinian groups with torsion we prove that every finitely generated kleinian group that contains a finite non cyclic subgroup either is finite or virtually free or contains a surface subgroup hence every arithmetic kleinian group contains a surface subgroup

increasing the number of fibered faces of arithmetic hyperbolic 3 manifolds we exhibit a closed hyperbolic 3 manifold which satisfies a very strong form of thurston s virtual fibration conjecture in particular this manifold has finite covers which fiber over the circle in arbitrarily many ways more precisely it has a tower of finite covers where the number of fibered faces of the thurston norm ball goes to infinity in fact faster than any power of the logarithm of the degree of the cover and we give a more precise quantitative lower bound the example manifold m is arithmetic and the proof uses detailed number theoretic information at the level of the hecke eigenvalues to drive a geometric argument based on fried s dynamical characterization of the fibered faces the origin of the basic fibration m s 1 is the modular elliptic curve e x 0 49 which admits multiplication by the ring of integers of q 7 we first base change the holomorphic differential on e to a cusp form on gl 2 over k q 3 and then transfer over to a quaternion algebra d k ramified only at the primes above 7 the fundamental group of m is a quotient of the principal congruence subgroup of o d of level 7 to analyze the topological properties of m we use a new practical method for computing the thurston norm which is of independent interest we also give a noncompact finite volume hyperbolic 3 manifold with the same properties by using a direct topological argument

non perturbative tricritical exponents of trails ii exact enumerations on square and simple cubic lattices for pt i see ibid vol 21 p 773 1988 the authors present exact enumerations of trails self intersecting but non overlapping lattice walks tabulated according to their length l number of intersection i and end to end distance for the square lattice up to l 21 i 7 and for the simple cubic lattice up to l 15 i 5 they introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach the existence of tricritical points in both lattices is manifested by the divergence of the specific heat the values of the tricritical couplings and exponents are extracted by a dlogpade analysis they find for the three dimensional lattice nu t approximately 0 48 and gamma t approximately 0 43 nu t approximately 0 52 and gamma t approximately 1 25 for the square lattice

on limit multiplicites of discrete series representations in spaces of automorphic forms of course the groups f are then arithmetic subgroups of g in this situation it seems natural to assume that as n the spectral decomposition of l2 f g will approximate that of g in particular let 3 be a discrete series representation of g if f is a subgroup of g let re f 6 be the multiplicity of 3 in the representation of g by right translations on l2 f g choose a haar measure dg on g if f is arithmetic let v f g be the finite volume of f g for the measure induced by dg let d 3 be the formal degree of 3 for the measure dg

intersection numbers of special cycles let g q be a connected semisimple algebraic group defined over q and let a r be two q rational automorphisms of finite order of g which commute with each other we assume that the group g r of real points of g is noncompact let i be a torsionfree arithmetic subgroup of g q which is stable under the group a r generated by a and r then a and r act on the space x of maximal compact subgroups of g r and on the locally symmetric space x i in a natural way if g f l denotes the reductive algebraic q group of fixed points of f l a r acting on g then the corresponding symmetric space x f l is acted upon by the arithmetic group 1 f l of fixed points of f l acting on i these give rise to nonempty closed immersed submanifolds c a x a 1 a and c r x r 1 r of x i they are called special cycles we assume that x i c a and c r are oriented and that the intersection c a n c r is compact then if dim c a dim c r dim x the intersection number

representations des groupes reductifs p adiques et de leurs sous groupes distingues cocompacts resume soient g et h deux groupes reductifs p adiques et h g une isogenie centrale a silberger a prouve qu une representation complexe admissible irreductible de g a pour restriction via une representation admissible semi simple de longueur finie de h nous prouvons que reciproquement toute representation complexe admissible irreductible de h est facteur direct d une telle restriction et s etend en une representation admissible d un sous groupe ouvert d indice fini de g nous etendons la portee de ces resultats en les reliant aux proprietes algebriques des categories de representations lisses de g et h let g h be reductive p adic groups and h g a central isogeny a silberger has proved that any admissible irreducible complex representation of g restricts via to a direct sum of finitely many admissible irreducible representations of h we prove that conversely any admissible irreducible complex representation of h occurs as a direct summand of such a restriction and extends to an admissible representation of a finite index open subgroup of g we extend those results by linking them with algebraic properties of the categories of smooth representations of g and h

cycles with local coefficients for orthogonal groups and vector valued siegel modular forms the purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and fourier coefficients of siegel modular forms to the case where the cycles have local coefficients now the correspondence will involve vector valued siegel modular forms

on the cuspidal cohomology of arithmetic subgroups of mathrm sl 2n and the first betti number of arithmetic 3 manifolds

appendix to the paper some uniserial representations of certain special linear groups by p sin and j g thompson abstract this appendix collects some material on the cohomology of discrete subgroups of lie groups and the theory of their representation varieties to provide background for the results of sin and thompson 2010 2013 in 22 23

introducing totally nonparallel immersions abstract an immersion of a smooth n dimensional manifold m r q is called totally nonparallel if for every distinct x y m the tangent spaces at f x and f y contain no parallel lines equivalently they span a 2n dimensional space given a manifold m we seek the minimum dimension tn m such that there exists a totally nonparallel immersion m r tn m in analogy with the totally skew embeddings studied by ghomi and tabachnikov we find that totally nonparallel immersions are related to the generalized vector field problem the immersion and embedding problems for real projective spaces and nonsingular symmetric bilinear maps our study of totally nonparallel immersions follows a recent trend of studying conditions which manifest on the configuration space f k m of k tuples of distinct points of m for example k regular embeddings k skew embeddings k neighborly embeddings and several others typically a map satisfying one of these configuration space conditions induces some s k equivariant map on the configuration space f k m or on a bundle thereof and obstructions can be computed in the form of stiefel whitney classes however the existence problem for such conditions is relatively unstudied our main result is a whitney type theorem every smooth n manifold m admits a totally nonparallel immersion into r 4 n 1 one dimension less than given by genericity we begin by studying the local problem which requires a thorough understanding of the space of nonsingular symmetric bilinear maps after which the main theorem is established using the removal of singularities h principle technique due to gromov and eliashberg when combined with a recent non immersion theorem of davis we obtain the exact value tn r p n 4 n 1 when n is a power of 2 this is the first optimal dimension result for any closed manifold m besides s 1 for any of the recently studied configuration space conditions

eigenvalues of the laplacian the first betti number and the congruence subgroup problem theorem 1 1 let f be an arithmetic lattice in the real lie group so n 1 n 2 if n 7 then assume r does not come from the twisted forms 3 6d4 if n 3 and f comes from the units of a quaternion algebra over a number field l with a unique complex embedding then assume l has a subfield of index 2 then for some finite index congruence subgroup fo of r the commutator quotient is infinite equivalently 1 fo the first betti number of the manifold fo hn is nonzero when 1 so n 1 so n is the n dimensional hyperbolic space

doubly cuspidal cohomology for principal congruence subgroups of gl 3 z the cohomology of arithmetic groups is made up of two pieces the cuspidal and noncuspidal parts within the cuspidal cohomology is a subspacethe f cuspidal cohomology spanned by the classes that generate representations of the associated finite lie group which are cuspidal in the sense of finite lie group theory few concrete examples of f cuspidal cohomology have been computed geometrically outside the cases of rational rank 1 or where the symmetric space has a hermitian structure this paper presents new computations of the f cuspidal cohomology of principal congruence subgroups f p of gl 3 2 of prime level p we show that the f cuspidal cohomology of f p vanishes for all p 19 with p 11 but that it is nonzero for p 11 we give a precise description of the f cuspidal cohomology for jt 11 in terms of the f cuspidal representations of the finite lie group gl 3 e 11 we obtained the result ultimately by proving that a certain large complex matrix m is rank deficient computation with the svd algorithm gave strong evidence that m was rank deficient but to prove it we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields

real space renormalization group approach to the theta point of a linear polymer in 2 and 3 dimensions abstract we present a real space renormalization group rsrg approach to a model of self avoiding walks saw with attractive interactions which is expected to describe the point transition of linear polymers both in two and in three dimensions the results of our approximations are physically sound and qualitatively consistent with other numerical predictions and theoretical conjectures in particular the collapsed phase is correctly described in the present approach

topological obstructions to totally skew embeddings following ghomi and tabachnikov s 2008 work we study the invariant n m n defined as the smallest dimension n such that there exists a totally skew embedding of a smooth manifold m n in r n this problem is naturally related to the question of estimating the geometric dimension of the stable normal bundle of the configuration space f 2 m n of ordered pairs of distinct points in m n we demonstrate that in a number of interesting cases the lower bounds on n m n obtained by this method are quite accurate and very close to the best known general upper bound n m n 1 n m n 4n 2 alpha n 1