Enumerating limit groups

We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (à la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of recognition algorithms for limit groups and free groups. A limit group is a finitely generated, fully residually free group. Recent research into limit groups has been motivated by their role in the theory of the set of homomorphisms from a finitely presented group to a free group, and in the logic of free groups. This research has culminated in the independent solutions to Tarski’s problems on the elementary theory of free groups by Z. Sela (see [21], [22] et seq.) and O. Kharlampovich and A. Miasnikov (see [12], [13] et seq.). Sela’s work extends to the elementary theory of hyperbolic groups [19]. We will be entirely concerned with finitely presentable groups. A class of groups G is recursively enumerable if there exists a Turing machine that outputs a list of presentations for every group G. T. Delzant asked if the class of limit groups is recursively enumerable [20, I.13]. Theorem A (Corollary 3.7). The class of limit groups is recursively enumerable. Addendum. In [4] and [8, 7] it is shown that the isomorphism problem is solvable for the class of limit groups. Therefore, one can improve the above