** Amenable simple groups **

joint with Nicolas Monod

I describe an example of a finitely generated infinite simple group. is amenable and simple, but not finitely generated).

**1. Construction **

Start with a Cantor space , a minimal self homeomorphism of . The *topological full group* is the group of all homeos of such that for all , there exists such that .

**Theorem 1 (Matui 2006)** * If is a minimal subshift (there exist uncounably many of them), the commutator subgroup of is simple and finitely generated. *

**2. Proof of amenability **

We view as a group of piecewise-translations of , i.e. such that is bounded. Indeed, orbits are dense, so it suffices to look at element s of acting on one single orbit, i.e. on . The group of piecewise-translations contains (van Douwen).

** 2.1. Our strategy **

1. We let act in a clever manner on finite subsets of , which is a group.

2. The action of on has locally finite (and thus amenable) stabilizers.

3. There exists a -invariant measure on . This suffices to prove that is amenable.

** 2.2. Action of **

We embed into the semi-direct product as follows: is mapped to . This yields the following action of on : .

** 2.3. Amenability of stabilizers **

Now is minimal action of on has *ubiquitous pattern property*, which means that for every finite set of elements , on a large interval of for a suitable picked nearby any point of .

We show that this implies that stabilizers are amenable.

To get the invariant mean on , it suffices to construct a sequence of -almost invariant functions in . We set

These are a modification of functions used by Kechris and Tsankov for the standard -action on . They used characteristic functions of sets ; on takes value 1 more often than 0.

Getting estimates on Folner function seems hopeless. Study of is interesting too.

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## About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See
http://www.math.ens.fr/metric2011/