# SUBGROUP DYNAMICS AND C∗-SIMPLICITY OF GROUPS OF HOMEOMORPHISMS

@inproceedings{Boudec2016SUBGROUPDA, title={SUBGROUP DYNAMICS AND C∗-SIMPLICITY OF GROUPS OF HOMEOMORPHISMS}, author={Adrien Le Boudec}, year={2016} }

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a C∗-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson’s group V is C∗-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented C∗-simple groups…

## 40 Citations

### Groups of piecewise linear homeomorphisms of flows

- MathematicsCompositio Mathematica
- 2020

To every dynamical system $(X,\varphi )$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of…

### The ideal intersection property for essential groupoid C*-algebras

- Mathematics
- 2021

We characterise, in several complementary ways, étale groupoids with locally compact Hausdorff space of units whose essential groupoid C∗-algebra has the ideal intersection property, assuming that…

### Rigidity properties of full groups of pseudogroups over the Cantor set

- Mathematics
- 2018

We show that the (topological) full group of a minimal pseudogroup over the Cantor set satisfies various rigidity phenomena of topological dynamical and combinatorial nature.
Our main result applies…

### On rigid stabilizers and invariant random subgroups of groups of homeomorphisms

- Mathematics
- 2019

A generalization of the double commutator lemma for normal subgroups is shown for invariant random subgroups of a countable group acting faithfully on a Hausdorff space. As an application, we…

### Piecewise strongly proximal actions, free boundaries and the Neretin groups

- Mathematics
- 2021

A closed subgroup H of a locally compact group G is confined if the closure of the conjugacy class of H in the Chabauty space of G does not contain the trivial subgroup. We establish a dynamical…

### Dynamical alternating groups, stability, property Gamma, and inner amenability

- Mathematics
- 2019

We prove that the alternating group of a topologically free action of a countably infinite group $\Gamma$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the…

### A commutator lemma for confined subgroups and applications to groups acting on rooted trees

- Mathematics
- 2020

A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a…

### Realizing uniformly recurrent subgroups

- MathematicsErgodic Theory and Dynamical Systems
- 2020

We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the…

### C * -simplicity of HNN extensions and groups acting on trees

- Mathematics
- 2017

We study groups admitting extreme boundary actions, and in particular, groups acting on trees, and we give necessary and sufficient criteria for such groups to be C*-simple or have the unique trace…

### Groups of Automorphisms and Almost Automorphisms of Trees: Subgroups and Dynamics

- Mathematics
- 2018

These are notes of a lecture series delivered during the program Winter of Disconnectedness in Newcastle, Australia, 2016. The exposition is on several families of groups acting on trees by…

## References

SHOWING 1-10 OF 37 REFERENCES

### Topological dynamics and group theory

- Mathematics
- 1974

We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a…

### Extensions of amenable groups by recurrent groupoids

- Mathematics
- 2016

We show that the amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class…

### Uniformly recurrent subgroups

- Mathematics
- 2014

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in a work of M. Abert, Y. Glasner and B.…

### On minimal, strongly proximal actions of locally compact groups

- Mathematics
- 2003

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known asboundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of…

### Subshifts with slow complexity and simple groups with the Liouville property

- Mathematics
- 2014

We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod…

### Invariant random subgroups of linear groups

- Mathematics
- 2014

AbstractLet Γ < GLn(F) be a countable non-amenable linear group with a simple, center free Zariski closure. Let Sub(Γ) denote the space of all subgroups of Γ with the compact, metric, Chabauty…

### C*-simplicity and the unique trace property for discrete groups

- Mathematics
- 2014

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical…

### Ideal structure of the C -algebra of Thompson group T

- Mathematics
- 2014

In a recent paper Ue Haagerup and Kristian Knudsen Olesen show that for Richard Thompson’s group T, if there exists a finite set H which can be decomposed as disjoint union of sets H1 and H2 with P…

### Boundaries of reduced C*-algebras of discrete groups

- Mathematics
- 2014

For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up…