Vol. 263, No. 1, 2013

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Fixed points of endomorphisms of virtually free groups

Pedro V. Silva

Vol. 263 (2013), No. 1, 207–240
Abstract

A fixed point theorem is proved for inverse transducers, which leads to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regular fixed points in the hyperbolic boundary has finitely many orbits under the action of the finite fixed points. In the automorphism case, it is shown that these regular fixed points are either exponentially stable attractors or exponentially stable repellers.

Keywords
virtually free groups, endomorphisms, fixed points, hyperbolic boundary, classification of fixed points
Mathematical Subject Classification 2010
Primary: 20E05, 20E36, 20F67
Secondary: 68Q45, 37B25
Milestones
Received: 12 April 2012
Revised: 7 November 2012
Accepted: 19 November 2012
Published: 14 May 2013
Authors
Pedro V. Silva
Centro de Matemática, Faculdade de Ciências
Universidade do Porto
R. Campo Alegre 687
4169-007 Porto
Portugal