# A new method for showing groups are amenable

Extensions of amenable groups by recurrent groupoids

Kate Juschenko (Vanderbilt), Volodymyr Nekrashevych (Texas A&M), Mikael De La Salle (ENS de Lyon)

This paper introduces a new framework to prove that certain groups are amenable.  There is an important distinction to make when talking about amenable groups: there are the elementary amenable groups, which are roughly speaking the groups which are obviously amenable, and then there are the nonelementary amenable groups.  It took about 25 years to show that nonelementary amenable groups exist, with the first example due to Grigorchuk.  It’s a very nice group, which can be defined in terms of measure-preserving transformations of $[0,1]$, or automorphisms of the complete binary tree, or even in terms of finite automata.  Other examples of nonelementary amenable groups followed, with various methods used to prove their amenability.  The main theorem of this paper can be used to show that any previously-known (to me, at least) example of a nonelementary amenable group is amenable.  That is, this one theorem implies the amenability of several different groups.  It also implies the amenability of groups not previously known to be amenable.

The main theorem is actually somewhat involved to state.  It says that if you have groups $G$ and $H$ of a certain type, and they satisfy four or five different conditions between them, then you can conclude that $G$ and the topological full group $[[G]]$ are amenable.

I’ve decided to do something a little different.  (What the heck, it’s the summer.)  I’m going to go through this paper and post a little bit as I do.  I’m not quite sure what I’ll write; it’s a bit silly to just reproduce the proofs in the paper, but it will be nice to sketch out the high points at least.  Tentatively: my next post will cover some of the background material used, the one after that will cover the main theorem, and the last post will show how various examples satisfy the conditions of the main theorem.  Let’s see how it goes.