Amenability of groups acting on trees

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation $$\Delta = < b,t|[b,t^2]b^{-1},[[[b,t^{-1}],b],b]>.$$ I show that it acts transitively on a 3-regular tree, and that $\Gamma=< b,b^{t^{-1}}$ stabilizes a vertex and acts by restriction on a binary rooted tree. $\Gamma$ is a "fractal group", generated by a 3-state automaton, and is a discrete analogue of the monodromy action of iterates of f(z)=z^2-1 on associated coverings of the Riemann sphere. $\Delta$ shares many properties with the Thompson group $F$. The proof of the main result (amenability of $\Delta$) is incomplete in the present form; please refer to the paper arxiv.org/math.GR/0305262, joint with Balint Virag, for a complete proof.