Conjugacy Separating Representations of Free Groups

Abstract

If $G$ is a free group and $g$ is an element of $G$ we show that there exists a residually finite (commutative) integral domain $R$ and a faithful matrix representation $\rho$ of $G$ over $R$ of finite degree such that the conjugacy class of $g\rho$ in $G\rho$ is closed in the topology induced on $G\rho$ by the Zariski topology on the full matrix algebra. It follows that free groups are conjugacy separable, a result obtained by a number of authors, see [1], [5] and [6].