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.

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/
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