## Notes of Kate Jushenko’s lecture

Amenable simple groups

joint with Nicolas Monod

I describe an example of a finitely generated infinite simple group.${(A(\infty)}$ is amenable and simple, but not finitely generated).

1. Construction

Start with a Cantor space ${C}$, ${T}$ a minimal self homeomorphism of ${C}$. The topological full group ${[[T]]}$ is the group of all homeos ${\phi}$ of ${C}$ such that for all ${x \in C}$, there exists ${n(x)}$ such that ${\phi(x)=T^n(x) (x)}$.

Theorem 1 (Matui 2006) If ${T}$ is a minimal subshift (there exist uncounably many of them), the commutator subgroup of ${[[T]]}$ is simple and finitely generated.

2. Proof of amenability

We view ${[[T]]}$ as a group of piecewise-translations of ${{\mathbb Z}}$, i.e. ${g:{\mathbb Z} \rightarrow {\mathbb Z}}$ such that ${|g(j)-j|}$ is bounded. Indeed, orbits are dense, so it suffices to look at element s of ${[[T]]}$ acting on one single orbit, i.e. on ${{\mathbb Z}}$. The group ${W({\mathbb Z})}$ of piecewise-translations contains ${F_2}$ (van Douwen).

2.1. Our strategy

1. We let ${W({\mathbb Z})}$ act in a clever manner on ${P(Z)=\{}$finite subsets of ${{\mathbb Z}\}}$, which is a group.

2. The action of ${[[T]]}$ on ${P({\mathbb Z})}$ has locally finite (and thus amenable) stabilizers.

3. There exists a ${W({\mathbb Z})}$-invariant measure on ${P(f)}$. This suffices to prove that ${[[T]]}$ is amenable.

2.2. Action of ${W({\mathbb Z})}$

We embed ${W({\mathbb Z})}$ into the semi-direct product ${P({\mathbb Z})\times W({\mathbb Z})}$ as follows: ${g}$ is mapped to ${(gN+N,g)}$. This yields the following action of ${W({\mathbb Z})}$ on ${P({\mathbb Z})}$: ${g(E)=N+g(N)+g(E)}$.

2.3. Amenability of stabilizers

Now ${T}$ is minimal ${\Leftrightarrow}$ action of ${[[T]]}$ on ${P({\mathbb Z})}$ has ubiquitous pattern property, which means that for every finite set of elements ${g}$, ${g(i+t)=g(i)+t}$ on a large interval of ${i}$ for a suitable ${t}$ picked nearby any point of ${{\mathbb Z}}$.

We show that this implies that stabilizers are amenable.

To get the invariant mean on ${P({\mathbb Z})}$, it suffices to construct a sequence of ${W({\mathbb Z})}$-almost invariant functions in ${L^2({0,1}^{\mathbb Z})}$. We set

$\displaystyle \begin{array}{rcl} f_n(\omega)=\exp(-n(\sum_j \omega_j \exp(-|j|/n))). \end{array}$

These are a modification of functions used by Kechris and Tsankov for the standard ${W({\mathbb Z})}$-action on ${P({\mathbb Z})}$. They used characteristic functions of sets ${A_n =\{\omega}$ ; on ${[-n,n],\, \omega}$ takes value 1 more often than 0${\}}$.

Getting estimates on Folner function seems hopeless. Study of ${W({\mathbb Z}^d)}$ is interesting too.