The twisted conjugacy problem for finitely generated free groups

Let G be a finitely generated free group and let phi is an element of End(G) be an endomorphism of G. In this paper we prove that the twisted conjugacy problem for phi is algorithmically solvable in the special case of phi having remnant. This case covers a significant set of endomorphisms. It is proved in Wagner (1999) [14] that almost all endomorphisms of G have remnant in a sense that can be made precise in terms of probability. For phi is an element of End(G) having remnant, we provide an upper bound on the length of elements z is an element of G that need to be checked to solve the twisted conjugate problem for phi so that the algorithm is simple to use for a computer search. Our new algorithm improves on existing algorithms which can only handle homomorphisms with remnant words of length at least 2. (C) 2015 Elsevier B.V. All rights reserved.