Orbit decidability and the conjugacy problem for some extensions of groups

Abstract:

Given a short exact sequence of groups with certain conditions, $1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb {Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb {Z}$, and $\mathbb {Z}^n \rtimes _A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb {Z})$. Also, we give an easy way of constructing groups of the form $\mathbb {Z}^4\rtimes F_n$ and $F_3\rtimes F_n$ 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 $Aut(F_2)$ is given.