# The isomorphism problem for hyperbolic groups I

### Abstract

Using canonical representatives in hyperbolic groups and the decidability of the Diophantine theory of free semigroups with paired alphabet, we solve the isomorphism problem for hyperbolic groups with no (essential) small action on a real tree. The solution enables computation of the outer automorphism group of such groups and implies the homeomorphism problem for “weakly geometric” 3-manifolds, closed hyperbolic manifolds, and negatively curved ones of dimension $\ge 5$. In a continuation paper [Sel], we combine structural results on hyperbolic groups and their automorphisms obtained in [Se2] with the procedure described in this paper to give a complete solution to the isomorphism problem for (torsion-free) hyperbolic groups.

Zlil Sela