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Groups, Geometry, and Dynamics

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Enumerating limit groups

Daniel Groves (1) and Henry Wilton (2)

(1) Department of Mathematics, Statistics & Computer S, University of Illinois at Chicago, 851 S. Morgan St., IL 60607-7045, CHICAGO, UNITED STATES
(2) Mathematics 253-37, California Institute of Technology, CA 91125, PASADENA, UNITED STATES

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.

Keywords: Limit groups, algorithmic properties