Orbit decidability and the conjugacy problem for some extensions of groups

Abstract

Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1 1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1 , we prove that G G has solvable conjugacy problem if and only if the corresponding action subgroup A ⩽ A u t ( F ) A\leqslant 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 \mathbb {Z}^2\rtimes F_m , F 2 ⋊ F m F_2\rtimes F_m , F n ⋊ Z F_n \rtimes \mathbb {Z} , and Z n ⋊ A F m \mathbb {Z}^n \rtimes _A F_m with virtually solvable action group A ⩽ G L n ( Z ) A\leqslant GL_n(\mathbb {Z}) . Also, we give an easy way of constructing groups of the form Z 4 ⋊ F n \mathbb {Z}^4\rtimes F_n and F 3 ⋊ F n F_3\rtimes 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 A u t ( F 2 ) Aut(F_2) is given.