69 research outputs found
Homotopy type and volume of locally symmetric manifolds
We consider locally symmetric manifolds with a fixed universal covering, and
construct for each such manifold M a simplicial complex R whose size is
proportional to the volume of M. When M is non-compact, R is homotopically
equivalent to M, while when M is compact, R is homotopically equivalent to M\N,
where N is a finite union of submanifolds of fairly smaller dimensions. This
expresses how the volume controls the topological structure of M, and yields
concrete bounds for various finiteness statements which previously had no
quantitative proofs. For example, it gives an explicit upper bound for the
possible number of locally symmetric manifolds of volume bounded by v>0, and it
yields an estimate for the size of a minimal presentation for the fundamental
group of a manifold in terms of its volume. It also yields a number of new
finiteness results.Comment: 54 pages, amsc
Limits of finite homogeneous metric spaces
We classify the metric spaces that can be approximated by finite homogeneous
ones.Comment: 9 page
Convergence groups are not invariably generated
It was conjectured in [KLS14] that non-elementary word hyperbolic groups are
never invariably generated. We show that this is indeed the case even for the
much larger class of convergence groups.Comment: 7 page
On fixed points and uniformly convex spaces
The purpose of this note is to present two elementary, but useful, facts
concerning actions on uniformly convex spaces. We demonstrate how each of them
can be used in an alternative proof of the triviality of the first
-cohomology of higher rank simple Lie groups, proved in [BFGM].Comment: 2 page
A lecture on Invariant Random Subgroups
Invariant random subgroups (IRS) are conjugacy invariant probability measures
on the space of subgroups in a given group G. They can be regarded both as a
generalization of normal subgroups as well as a generalization of lattices. As
such, it is intriguing to extend results from the theories of normal subgroups
and of lattices to the context of IRS. Another approach is to analyse and then
use the space IRS(G) as a compact G-space in order to establish new results
about lattices. The second approach has been taken in the work [7s12], that
came to be known as the seven samurai paper. In these lecture notes we shall
try to give a taste of both approaches.Comment: 19 pages, 5 figures. Based on a mini course given in Ghys' birthday
--- conference Geometries in Action (2015), Oberwolfach workshop on Locally
Compact Groups (2014) and Ventotene's conference on Manifolds and Groups
(2015
Kazhdan-Margulis theorem for Invariant Random Subgroups
Given a simple Lie group , we show that the lattices in are weakly
uniformly discrete. This is a strengthening of the Kazhdan-Margulis theorem.
Our proof however is straightforward --- considering general IRS rather than
lattices allows us to apply a compactness argument. In terms of p.m.p. actions,
we show that for every there is an identity neighbourhood which
intersects trivially the stabilizers of of the points in every
non-atomic -space.Comment: 4 page
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.Comment: 8 page
The dynamics of Aut(F_n) on redundant representations
We study some dynamical properties of the canonical Aut(F_n)-action on the
space R_n(G) of redundant representations of the free group F_n in G, where G
is the group of rational points of a simple algebraic group over a local field.
We show that this action is always minimal and ergodic, confirming a conjecture
of A. Lubotzky. On the other hand for the classical cases where G=SL(2,R) or
SL(2,C) we show that the action is not weak mixing, in the sense that the
diagonal action on R_n(G)^2 is not ergodic.Comment: Some of the statements and arguments rely on the assumption that the
algebraic group G is simply connected. This assumption, which was missing in
the previous version, is not necessary in the archimedean cases, but it is
needed in the non-archimedean case
A topological Tits alternative
Let k be a local field, and G a linear group over k. We prove that either G
contains a relatively open solvable subgroup, or it contains a relatively dense
free subgroup. This result has applications in dynamics, Riemannian foliations
and profinite groups.Comment: 43 page
Equicontinuous actions of semisimple groups
We study equicontinuous actions of semisimple groups and some
generalizations. We prove that any such action is universally closed, and in
particular proper. We derive various applications, both old and new, including
closedness of continuous homomorphisms, nonexistence of weaker topologies,
metric ergodicity of transitive actions and vanishing of matrix coefficients
for reflexive (more generally: WAP) representations.Comment: 28 pages. The introduction has been improved. (We also extended the
discussion about week topologies.
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