Finite groups and their actions on surfaces

Let G be a finite group and let ^ be a compact, connected and orientable surface with genus g. In this study, we work on when G, which has faithful action, was given as a finite group of ^ orientation-preserving self-homeomorphisms, we determine ^"(G) set of £ values. There exist a positive integer K, depending only on the order d = \G\ of G, the exponent e = expG of G, and the structure of a Sylow 2- subgroup G2 of G, satisfying: Ç(G) consists of all but finitely many non-negative integers g = 1 (mod AT). Furthermore, the set f(G) of g such that G acts freely on ^, is obtained. In this case, ]T -> ^ I G is unbranched. We can obtain this set by using Cayley graph of G. '.... '<:? 'W3 } 3 V'.:- ': *?..p.J. ***? 11 Sk^..,.'*