Homogeneity in torsion-free hyperbolic groups

In his answer to Tarski’s problem on the elementary equivalence of free groups of different ranks, Sela developed a number of geometric tools to understand the first-order theory of free groups. The geometric nature of these techniques enabled him to extend his results to torsion-free hyperbolic groups, a class of finitely presented groups which are in many ways close to free groups, but also in some sense “generic” among finitely presented groups. They contain in particular free groups and fundamental groups of (sufficiently large) surface groups.

Using some of Sela’s tools, we proved together with Sklinos that free groups are homogeneous (there is also an independent proof by Ould Houcine), but that the fundamental groups of most closed surfaces are not. Together with Ayala Byron, we give criterions of an algebraic nature to determine whether a given torsion-free hyperbolic group is homogeneous.

In the talk we will give an overview of known results about homogeneity of various groups, introduce the class of hyperbolic groups and explain how to determine whether such a group is homogeneous.