Integral Congruence Two Hyperbolic 5-Manifolds

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form H ⁵/ where is a torsion-free subgroup of minimal index of the congruence two subgroup ⁵ ₂ of the group ⁵ of positive units of the Lorentzian quadratic form x 2/1 +... +x 5/2 -x 6/2. We also show that ⁵ ₂ is a reflection group with respect to a 5-dimensional right-angled convex polytope in H ⁵. As an application, we construct a hyperbolic 5-manifold of smallest known volume 7 (3)/4.     

The thinnest holding-lattice of a set

In the present note we are going to study the following question proposed by L. Fejes Toth. In the Euclidean plane a lattice is called a holding-lattice of a planar setS if any set congruent toS contains at least one lattice point of . The density of equals (2)^–1, where denotes the area of a fundamental triangle of . A holding-lattice ofS of least possible density is said to be the thinnest holding-lattice ofS. The problem: Find the sets whose thinnest holding-lattice is a regular trianglelattice.To Professor L. Fejes Toth on his 70th Birthday     

Bounding the diameter of a distance regular graph by a function ofk d

Let be a distance regular graph with diameterd, and _d () the set of vertices at distanced from. is said to be thin if the induced subgraph on _d () is a union of cliques for every vertex. We show that the diameterd is bounded above by a function depending only onk _d, which is the cardinality of _d (), if is not thin. We also investigate thin distance regular graphs witha _d 0.     

Mortality of iterated Gallai graphs

For a finite or infinite graphG, theGallai graph (G) ofG is defined as the graph whose vertex set is the edge setE(G) ofG; two distinct edges ofG are adjacent in (G) if they are incident but do not span a triangle inG. For any positive integert, thetth iterated Gallai graph ^t (G) ofG is defined by ( ^t–1(G)), where ⁰(G):=G. A graph is said to beGallai-mortal if some of its iterated Gallai graphs finally equals the empty graph. In this paper we characterize Gallai-mortal graphs in several ways.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

Reconstruction of a submanifold of Euclidean space from its Grassmannian image that degenerates into a line

We study the existence of a submanifoldF ⁿ of Euclidean spaceE ^n+p with prescribed Grassmannian image that degenerates into a line. We prove that is the Grassmannian image of a regular submanifoldF ⁿ of Euclidean spaceE ^n+p if and only if the curve in the Grassmann manifoldG ⁺ (p, n+p) is asymptoticallyC ^r -regular,r>1. HereG ⁺ (n, n+p) is embedded into the sphereS ^N ,N=C _n+p ^p =( _p ^n+p ), by the Plücker coordinates.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 681–691, May, 1996.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Cohomology of Compact Locally Symmetric Spaces

We obtain a necessary condition for a cohomology class on a compact locally symmetric space S()=X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S _H()=Y of X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ⁿ and ^m , then low degree cohomology classes on the variety S() restrict non-trivially to the subvariety S _H (); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S().     

K-Theory for C*-Algebras of One-Relator Groups

We compute the K-theory groups of the reduced C*-algebra C _r ^* () of a one-relator group . We prove that every such group is K-amenable in the sense of Cuntz. For a torsion-free one-relator group =<X|r> such that r is not a product of commutators, we give a direct proof of the fact that the Baum–Connes analytical assembly map

Splitting with Continuous Control in Algebraic K-Theory

In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups that satisfy certain geometric conditions. The group is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K₀(k) is proved to be isomorphic to the colimit of K₀(kH) over the finite subgroups H of , when is a virtually polycyclic group and k is a field of characteristic zero.     

Tiling the torus and other space forms

We consider graphs on two-dimensional space forms which are quotient graphs /F, where is an infinite, 3-connected, face, vertex, or edge transitive planar graph andF is a subgroup of Aut(), all of whose elements act freely on . The enumeration of quotient graphs with transitivity properties reduces to computing the normalizers in Aut() of the subgroupsF. Results include: all isogonal toriodal polyhedra belong to the two families found by Grünbaum and Shephard; there are no transitive graphs on the Möbius band; there is a graph on the Klein bottle whose automorphism group acts transitively on its faces, edges, and vertices.This paper is an expanded version of a lecture presented to the Conference on Combinatorial Geometry, Oberwolfach, Germany, September 1984.     

Extended generalized octagons and the groupHe

Let be an extended generalized octagon such that the points of a triple {u, v, w} not on a block are pairwise adjacent if and only if the distance betweenv andw in the local generalized octagon _u equals 3 and there is a thick line through any point of _u . Then is one of the two examples related to the groups 2·L ₃(4).2² andHe. It is also shown that does not admit further extensions.     

Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995     

Bicoset groups and symmetric graphs

In this paper, we construct a functor from the set of bicoset multivalued groups with one Hermitian generator to the set of symmetric graphs. The functor makes it possible to describe the set of bicoset multivalued groups with one Hermitian generator as a sum of categories. The categories are indexed by pairs (^U,G^U), where ^U is a universal symmetric graph and G^U is a subgroup of Aut(^U). Bibliography: 5 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 161–174.This revised version was published online in April 2005 with a corrected cover date and article title.     

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space H ⁿ⁺¹. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = \H ⁿ⁺¹, under the assumption that H ¹(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.     

On the Neumann-Poincaré operator

Let be a rectifiable Jordan curve in the finite complex plane which is regular in the sense of Ahlfors and David. Denote by L _C ² () the space of all complex-valued functions on which are square integrable w.r. to the arc-length on . Let L ²() stand for the space of all real-valued functions in L _C ² () and put Since the Cauchy singular operator is bounded on L _C ² (), the Neumann-Poincaré operator C ₁ sending each h L ²() into , is bounded on L ²(). We show that the inclusion characterizes the circle in the class of all AD-regular Jordan curves .     

Convexity theories 0. Foundations

For an arbitrary prenormed semiring, the closed unit ball functor from the category _R pnSmod ¹ ofR-prenormedR-semimodules with contractions to the category of sets has a left adjoint. For such a semiringR the notion of finitary convexity theory overR is introduced and the category C of -modules is defined. It is shown that the canonical functor _R pnSmod ¹ C has a left adjoint. In caseR is a banach semiring one has infinitary convexity theories, in addition to the finitary ones, and again the canonical functor _R bnSmod ¹ C has a left adjoint.Many more happy returns, Nico. Sixty is forever.