# 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} }

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.

#### 27 Citations

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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 a… Expand

Locally compact groups whose ergodic or minimal actions are all free

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We construct locally compact groups with no non-trivial Invariant Random Subgroups and no non-trivial Uniformly Recurrent Subgroups.

The scale function and lattices

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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 restrictions… Expand

Conjugacy and Dynamics in Almost Automorphism Groups of Trees

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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 by… Expand

Groups acting on trees with almost prescribed local action

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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 generated… Expand

Non-discrete simple locally compact groups

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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 new… Expand

Compact presentability of tree almost automorphism groups

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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 include… Expand

A view on Invariant Random Subgroups and Lattices

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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… Expand

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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 of… Expand

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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-Solitar… Expand

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