Conjugacy separability of certain Fuchsian groups

Abstract:

Let $G$ be a group. An element $g$ is c.d. in $G$ if and only if given any element $h$ of $G$, either it is conjugate to $h$ or there is a homomorphism $\xi$ from $G$ onto a finite group such that $\xi (g)$ is not conjugate to $\xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. Let $F$ be a Fuchsian group, i.e. let $F$ be presented as \[ F = ({S_1}, \ldots ,{S_n},{a_1}, \ldots ,{a_{2r}},{b_1}, \ldots ,{b_t};S_{{1^1}}^e = \cdots = S_{{n^n}}^e = {S_1} \ldots {S_n}{a_1} \ldots {a_{2r}}a_1^{ - 1} \ldots a_{2r}^{ - 1}{b_1} \ldots {b_t} = 1).\] In this paper, we show that every element of infinite order in $F$ is c.d. and if $t \ne 0$ or $r \ne 0$, $F$ is c.s.