arXiv subjects
Publications and source records attributed to Tsachik Gelander.
We show that every dense subgroup of a connected Lie group G contains a dense subgroup generated by 2d elements, where d=dim(G). We also give a detailed proof for the quantitive characterization of a contracting projective transformation in terms of the ratio between the two leading terms in its Cartan decomposition.
We show that MIF (mixed-identity-free) linear groups are sharply MIF and linearly MIF. Along the way we provide a self contained proof of the strong approximation theorem, and a new (probabilistic) variant of the super approximation theorem.
In recent years various results about locally symmetric manifolds were proven using probabilistic approaches. One of the approaches is to consider random manifolds by associating a probability measure to the space of discrete subgroups of the isometry Lie group. The main goals are to prove results about deterministic groups and manifolds by considering appropriate measures. In this overview paper we describe several such results, observing the evolution process of the measures involved. Starting with a result whose proof considered finitely supported measures (more precisely, measures supported on finitely many conjugacy classes) and proceeding with results which were outcome of the successful and popular theory of IRS (invariant random subgroups). In the last couple of years the theory has expanded to SRS (stationary random subgroups) allowing to deal with a lot more problems and establish stronger results. In the last section we shall review a very recent (yet unpublished) result whose proof make use of random subgroups which are not even stationary.
We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis and removes the property (T) assumption from the recent counterpart result of Fraczyk and Gelander. We further show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is an irreducible lattice. This implies a variant of the Stuck-Zimmer conjecture under a strong irreducibility assumption of the action.
We give an elementary proof of a weak variant of the homotopy type conjecture from [Gel04]. The proof relies on Belolipetskys estimate on the systole in terms of the volume which is a consequence of Prasads volume formula. In particular we extend the estimate established in [B20] for hyperbolic manifolds to general arithmetic manifolds.
We prove the following conjecture of Margulis. Let $G$ be a higher rank simple Lie group and let $Λ\le G$ be a discrete subgroup of infinite covolume. Then, the locally symmetric space $Λ\backslash G/K$ admits injected balls of any radius. This can be considered as a geometric interpretation of the celebrated Margulis normal subgroup theorem. However, it applies to general discrete subgroups not necessarily associated to lattices. Yet, the result is new even for subgroups of infinite index of lattices. We establish similar results for higher rank semisimple groups with Kazhdan's property (T). We prove a stiffness result for discrete stationary random subgroups in higher rank semisimple groups and a stationary variant of the Stück-Zimmer theorem for higher rank semisimple groups with property (T). We also show that a stationary limit of a measure supported on discrete subgroups is almost surely discrete.
J. Wiegold conjectured that if n>2 and G is a finite simple group, then the action of Aut(F_n) on Epi(F_n,G) is transitive. In this note we consider analogous questions where G is a compact Lie group, a non-compact simple analytic group or a simple algebraic group.
This book is concerned with analytic approaches of studying groups and their actions. Much attention is devoted to the study of amenability and Kazhdan's property (T), which are perhaps the most important analytic properties of a group, but we also discuss other analytic notions. We tried to introduce tricks, ideas and lemmas that repeatedly turn out to be useful in various situations. Our main guideline was to expose the beauty of the theory and to present many different aspects of it while keeping the text short, simple and accessible, sometimes at the expense of diving deep or providing thorough expositions. Hopefully this book could serve as a smooth entry to Analytic Group Theory.
We show that an arithmetic lattice $Γ$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $δ(v)$ where $v = μ(G/Γ)$ is the co-volume of the lattice. We prove that $δ$ is polynomial in general and poly-logarithmic under GRH. We then show that this poly-logarithmic bound is almost optimal, by constructing certain lattices with torsion elements of order $\sim \frac{\log v}{\log \log v}$.
We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequence of finite volume $X$-manifolds, then the normalized Betti numbers $b_k(M_n)/vol(M_n)$ converge for all $k$. As a corollary, if $X$ has higher rank and $(M_n)$ is any sequence of distinct, finite volume $X$-manifolds, the normalized Betti numbers of $M_n$ converge to the $L^2$ Betti numbers of $X$. This extends our earlier work with Nikolov, Raimbault and Samet, where we proved the same convergence result for uniformly thick sequences of compact $X$-manifolds.
We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $βr^δ$ for some explicit constants $β, δ> 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.
It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated.
We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semi-simple case. In particular, we extend Mostow's theorem from solvable to amenable groups.
This paper is a survey on the works [MS77, MS79, MS81] on maximal subgroups in finitely generated linear groups, and the works that followed it [GG08, GG13b, GG13a, Kap03, Iva92, HO16, GM16, AGS14, Sf90, Sf98, Per05, AKT16, FG18, GS17] concerning maximal subgroups of infinite index in linear groups as well as in various other groups possessing a suitable geometry or dynamics.
Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension $d\ge 5$ up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest.
For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields.
We prove the Gaschütz Lemma holds for all metrisable compact groups.
We establish topological local rigidity for uniform lattices in compactly generated groups, extending the result of Weil from the realm of Lie groups. We generalize the classical local rigidity theorem of Selberg, Calabi and Weil to irreducible uniform lattices in $\text{Isom}(X)$ where $X$ is a proper $\text{CAT}(0)$ space with no Euclidian factors, not isometric to the hyperbolic plane. We deduce an analog of Wang's finiteness theorem for certain non-positively curved metric spaces.