Towers and the first-order theory of hyperbolic groups

This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$ relative to $H$ (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin's results just mentioned is true. This follows from the solution to Tarski's problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fra\"iss\'e's method, we associate to every torsion-free hyperbolic group $H$ a unique homogeneous countable group $\mathcal{M}$ in which any hyperbolic group $H'$ elementarily equivalent to $H$ has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group $H$ are $H$-limit groups.