# Simple groups without lattices

@article{Bader2012SimpleGW,
title={Simple groups without lattices},
author={U. Bader and P. Caprace and T. Gelander and S. Mozes},
journal={Bulletin of The London Mathematical Society},
year={2012},
volume={44},
pages={55-67}
}
• U. Bader, +1 author S. Mozes
• Published 2012
• Mathematics
• Bulletin of The London Mathematical Society
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.
Topological models of finite type for tree almost automorphism groups
• Mathematics
• 2015
We show that tree almost automorphism groups, including Neretin groups, satisfy the analogue of the $F_\infty$-finiteness condition in the world of totally disconnected groups: They possess aExpand
Locally compact groups whose ergodic or minimal actions are all free
• Mathematics
• 2017
We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups.
The scale function and lattices
It is shown that, given a lattice H in a totally disconnected, locally compact group G, the contraction subgroups in G and the values of the scale function on G are determined by their restrictionsExpand
Conjugacy and Dynamics in Almost Automorphism Groups of Trees
• Mathematics
• 2019
We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree byExpand
Groups acting on trees with almost prescribed local action
We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generatedExpand
Non-discrete simple locally compact groups
Simple Lie groups and simple algebraic groups over local fields are the most prominent members of the class S of compactly generated non-discrete simple locally compact groups. We outline a newExpand
Compact presentability of tree almost automorphism groups
We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results includeExpand
A view on Invariant Random Subgroups and Lattices
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 naturalExpand
L2‐Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices
• Mathematics
• 2016
The main result is a general approximation theorem for normalised Betti numbers for Farber sequences of lattices in totally disconnected groups. Further, we contribute to the general theory ofExpand
High transitivity for more groups acting on trees
• Mathematics
• 2020
We establish a new sharp sufficient condition for groups acting on trees to be highly transitive. This give new examples of highly transitive groups, including icc non-solvable Baumslag-SolitarExpand

#### References

SHOWING 1-10 OF 15 REFERENCES
On the order of doubly transitive permutation groups
AbstractOur aim is to contribute to an old problem of group theory. We prove that the order of a doubly transitive permutation group of degreen other thanAnorSnis less than exp exp (c\sqrt {\log n}Expand
Simple locally compact groups acting on trees and their germs of automorphisms
• Mathematics
• 2011
Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the globalExpand
On the orders of primitive groups
Abstract Almost all primitive permutation groups of degree n have order at most n· ∏ i=0 [ log 2 n]−1 (n−2 i ) 1+[ log 2 n] , or have socle isomorphic to a direct power of some alternating group. TheExpand
Finite Permutation Groups
1 Multiply transitive groups Theorem 1.1. Let Ω be a finite set and G ≤ Sym(Ω) be 2–transitive. Let N E G be a minimal normal subgroup. Then one of the following holds: (a) N is regular andExpand
Fourier Analysis on Groups
In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compactExpand
On the Order of Uniprimitive Permutation Groups
One of the central problems of 19th century group theory was the estimation of the order of a primitive permutation group G of degree n, where G X An. We prove I G I < exp (4V'/ n log2 n) for theExpand
Abstract Commensurators of Groups Acting on Rooted Trees
AbstractWe show that the abstract commensurator of a nearly level transitive weakly branch group H coincides with the relative commensurator of H in the homeomorphism group of the boundary of theExpand
ON COMBINATORIAL ANALOGS OF THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE
The goal of this article is to construct and study groups which, from the point of view of the theory of representations, should resemble the group of diffeomorphisms of the circle. The first type ofExpand
Discrete subgroups of Lie groups
Preliminaries.- I. Generalities on Lattices.- II. Lattices in Nilpotent Lie Groups.- III. Lattices in Solvable Lie Groups.- IV. Polycyclic Groups and Arithmeticity of Lattices in Solvable LieExpand
Compact Clifford-Klein forms of symmetric spaces
We recall that a Riemannian manifold X is symmetric, in the sense of Cartan, if it is connected and if every point x E X is an isolated fixed point of an involutive isometry s, of X. The map s, isExpand