Commensurator Subgroups of Surface Groups

let m be a surface, and let h be a subgroup of π1m. in this paper we study the commensurator subgroup c\\pi_1m(h) of π1m, and we extend a result of l. paris and d. rolfsen [7], when h is a geometric subgroup of π1m. 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.