$FC^-$-elements in totally disconnected groups and automorphisms of infinite graphs

Abstract

An element in a topological group is called an $\mathrm{FC}^-$-element if its conjugacy class has compact closure. The $\mathrm{FC}^-$-elements form a normal subgroup. In this note it is shown that in a compactly generated totally disconnected locally compact group this normal subgroup is closed. This result answers a question of Ghahramani, Runde and Willis. The proof uses a result of Trofimov about automorphism groups of graphs and a graph theoretical interpretation of the condition that the group is compactly generated.