Orbit decidability and the conjugacy problem for some extensions of groups

Abstract

Abstract. Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1, we prove that G has solvable conjugacy problem if and only if the corresponding action subgroup A Aut(F ) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z 2 F m , F 2 F m , F n Z, and Z n A F m with virtually solvable action group A GL n (Z). Also, we give an easy way of constructing groups of the form Z 4 F n and F 3 F n with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and we give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F 2 ) is given.

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

“…The first corollary shows that also the conjugacy problem of the Z 4 -byfree group of [2] (with undecidable conjugacy problem) is strongly generically decidable in polynomial time.…”

“…2 The distinction between elliptic and hyperbolic elements stems from group actions on trees. The group G acts naturally on a tree: its Bass-Serre tree corresponding to the splitting.…”

“…We present two more applications of our results: First, we show in Corollary 4 that the conjugacy problem of the Z 4 -by-free group from [2] -and, more generally, of all fundamental groups of finite graphs of groups with finitely generated free abelian vertex groups -is decidable in polynomial time on a strongly generic set.…”

A recursive presentation for Mihailova’s subgroup

“…In [2], O. Bogopolski, A. Martino and E. Ventura studied the conjugacy problem for extensions of groups. In that context, the notion of orbit decidability is crucial and we recall it here.…”

“…As a corollary we deduce the existence of a finitely generated, orbit undecidable subgroup of Aut.F 3 / (see [2] for details), which has the recursive presentation given in Theorem 1.1. Another application of Mihailova's construction can be found in [10], Proposition 5.4.…”