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Tsachik Gelander

Publications and source records attributed to Tsachik Gelander.

At least 55 records · Page 3Linked to original sources
arXiv2014

Counting commensurability classes of hyperbolic manifolds

Gromov and Piatetski-Shapiro proved existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about v^v such manifolds of volume at most v, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

math.GT↗
arXiv2014

Lectures on Lattices and locally symmetric spaces

The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated results which, in particular, shows some connections to other fields of mathematics. The time restriction forces us to avoid many important parts of the theory, and the route we have chosen is naturally biased by the individual taste of the speaker.

math.GR↗
arXiv2013

An Aschbacher--O'Nan--Scott theorem for countable linear groups

The purpose of this note is to extend the classical Aschbacher--O'Nan--Scott theorem for finite groups to the class of countable linear groups. This relies on the analysis of primitive actions carried out in a previous paper. Unlike the situation for finite groups, we show here that the number of primitive actions depends on the type: linear groups of almost simple type admit infinitely (and in fact unaccountably) many primitive actions, while affine and diagonal groups admit only one. The abundance of primitive permutation representations is particularly interesting for rigid groups such as simple and arithmetic ones.

math.GR↗
arXiv2012

Homogeneous number of free generators

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group Γwhich admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group Γis uncountable.

math.GR↗
arXiv2012

Milnor-Wood inequalities for products

We prove Milnor-Wood inequalities for local products of manifolds. As a consequence, we establish the generalized Chern Conjecture for products $M\times Σ^k$ for any product of a manifold $M$ with a product of $k$ copies of a surface $Σ$ for $k$ sufficiently large.

math.AT↗
arXiv2011

A fixed point theorem for L^1 spaces

We prove a fixed point theorem for a family of Banach spaces, notably L^1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the "derivation problem" studied since the 1960s.

math.FA↗
arXiv2011

Ice-Creams and Wedge Graphs

What is the minimum angle $α>0$ such that given any set of $α$-directional antennas (that is, antennas each of which can communicate along a wedge of angle $α$), one can always assign a direction to each antenna such that the resulting communication graph is connected? Here two antennas are connected by an edge if and only if each lies in the wedge assigned to the other. This problem was recently presented by Carmi, Katz, Lotker, and Rosén \cite{CKLR10} who also found the minimum such $α$ namely $α=\fracπ{3}$. In this paper we give a simple proof of this result. Moreover, we obtain a much stronger and optimal result (see Theorem \ref{theorem:main}) saying in particular that one can chose the directions of the antennas so that the communication graph has diameter $\le 4$. Our main tool is a surprisingly basic geometric lemma that is of independent interest. We show that for every compact convex set $S$ in the plane and every $0 < α< π$, there exist a point $O$ and two supporting lines to $S$ passing through $O$ and touching $S$ at two \emph{single points} $X$ and $Y$, respectively, such that $|OX|=|OY|$ and the angle between the two lines is $α$.

cs.CG↗
arXiv2011

On the growth of Betti numbers of locally symmetric spaces

We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gromov in \cite{BGS}. The basic idea is to adapt the theory of local convergence, originally introduced for sequences of graphs of bounded degree by Benjamimi and Schramm, to sequences of Riemannian manifolds. Using rigidity theory we are able to show that when the volume tends to infinity, the manifolds locally converge to the universal cover in a sufficiently strong manner that allows us to derive the convergence of the normalized Betti numbers.

math.GR↗
arXiv2011

Simple groups without lattices

We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups.

math.GR↗
arXiv2011

Compactifications and algebraic completions of Limit groups

In this paper we consider the existence of dense embeddings of Limit groups in locally compact groups generalizing earlier work of Breuillard, Gelander, Souto and Storm [GBSS] where surface groups were considered. Our main results are proved in the context of compact groups and algebraic groups over local fields. In addition we prove a generalization of the classical Baumslag lemma which is a useful tool for generating eventually faithful sequences of homomorphisms. The last section is dedicated to correct a mistake from [BGSS] and to get rid of the even genus assumption.

math.GR↗
arXiv2011

Volume vs. rank of lattices

We show the rank (i.e. minimal size of a generating set) of lattices cannot grow faster than the volume.

math.GR↗
arXiv2010

Counting arithmetic lattices and surfaces

We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $Γ$ as a function of its covolume), number theoretic (bounding the number of maximal such $Γ$) and sharp estimates on the character values of the symmetric groups (to bound the subgroup growth of $Γ$).

math.GR↗
arXiv2009

The generalized Chern conjecture for manifolds that are locally a product of surfaces

We consider closed manifolds that admit a metric locally isometric to a product of symmetric planes. For such manifolds, we prove that the Euler characteristic is an obstruction to the existence of flat structures, confirming an old conjecture proved by Milnor in dimension 2. In particular, the Chern conjecture follows in these cases. The proof goes via a new sharp Milnor--Wood inequality for Riemannian manifolds that are locally a product of hyperbolic planes. Furthermore, we analyze the possible flat vector bundles over such manifolds. Over closed Hilbert--Blumenthal modular varieties, we show that there are finitely many flat structures with nonzero Euler number and none of them corresponds to the tangent bundle. Some of the main results were announced in C.R. Acad. Sci. Paris, Ser. I 346 (2008) 661-666.

math.GT↗
arXiv2009

Dense embeddings of surface groups

We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dense faithfully embedded surface group. Similarly, we show that any connected semisimple Lie group contains a dense copy of any fully residually free group.

math.GR↗
arXiv2008

Milnor-Wood inequalities for manifolds locally isometric to a product of hyperbolic planes

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The manifolds under consideration are of particular interest, since in contrary to many other locally symmetric spaces they do admit flat vector bundle of the corresponding dimension. When the manifold is irreducible and of higher rank, it is shown that flat oriented vector bundles are determined completely by the sign of the Euler number.

math.GT↗
arXiv2006

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product.

math.GR↗