Commensurator Subgroups of Surface Groups

Let $M$ be a surface, and let $H$ be a subgroup of $\pi_{1}M$. In this paper we study the commensurator subgroup $C_{\pi_{1}M}(H)$ of $\pi_{1}M$, and we extend a result of L. Paris and D. Rolfsen \cite{Paris-Rolfsen}, when $H$ is a geometric subgroup of $\pi_{1}M$. We also give an application of commensurator subgroups to group representation theory. Finally, by considering certain closed curves on the Klein bottle, we apply a classification of these curves to self-intersection Nielsen theory.