Fixed points of endomorphisms of virtually free groups

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. Throughout the paper, the ambient groups are assumed to be finitely generated. Gersten [1987] proved that the fixed point subgroup of a free group automorphism ’ is finitely generated. Using a different approach, Cooper [1987] gave an alternative proof, proving also that the fixed points of the continuous extension of ’ to the boundary of the free group is, in some sense, finitely generated. Bestvina and Handel [1992] achieved a major breakthrough with their innovative train track techniques, bounding the rank of the fixed point subgroup and the generating set for the infinite fixed points. Their approach was pursued by Maslakova [2003], who considered the problem of effectively computing a basis for the fixed point subgroup. The paper turned out to contain some errors, and subsequently a new paper by Bogopolski and Maslakova [2012] was posted on arXiv with the purpose of correcting these errors. Gersten’s result was generalized to further classes of groups and endomorphisms in subsequent years. Goldstein and Turner extended it to monomorphisms of free groups [1985] and to arbitrary endomorphisms [1986]. Collins and Turner extended it to automorphisms of free products of freely indecomposable groups [1994]; The author acknowledges support from the European Regional Development Fund through the COMPETE program and from the Portuguese government through Fundacao para a Ciencia e a Tecnologia, under the project PEst-C/MAT/UI0144/2011. MSC2010: primary 20E05, 20E36, 20F67; secondary 68Q45, 37B25.