Elementary embeddings in torsion-free hyperbolic groups

Abstract

The purpose of this thesis is to describe first-order logic elementary embeddings in a torsion-free hyperbolic group. The main result gives such a description in terms of a structure that Sela introduced in his answer to Tarski's problem: the structure of hyperbolic tower. Thus, if H embedds elementarily in a torsion free hyperbolic group G, the group G can be obtained by successive amalgamations of groups of surfaces with boundary to a free product of H with some free group and groups of surfaces without boundary. This gives as a corollary that an elementary subgroup of a finitely generated free group is a free factor. The techniques used to obtain this description are mostly geometric, as for example actions on real or simplicial trees, or the existence of JSJ splittings. We also rely on the existence of factor sets, namely the fact that for some finitely generated groups A, given a torsion-free hyperbolic group G, there exists a finite set of quotients of A such that any non-injective morphism from A to G factors through one of the corresponding quotient maps up to precomposition by an automorphism of A. We give a full proof of these results, including a detailed version of Rips and Sela's shortening argument. The shortening argument shows, using Rips' analysis of actions on real trees, that if a sequence of actions of A on hyperbolic spaces converges to a real A-tree of a specific type, then infinitely many of the actions can be shortened.