Subgroups of Mod(S) generated by X in {(T_aT_b)^k,(T_bT_a)^k} and Y in {T_a,T_b}

Abstract: Suppose a and b are distinct isotopy classes of essential simple closed curves in an orientable surface S. Let T_a and T_b represent the respective Dehn twists along a and b. In this paper, we study the subgroups of Mod(S) generated by X and Y, where X belongs to {(T_aT_b)^k,(T_bT_a)^k}, k an integer, and Y belongs to {T_a,T_b}. For a large class of examples, we show that the subgroups <X,Y> and <T_a,T_b> are isomorphic. Moreover, we prove that <X,Y> = <T_a,T_b> whenever i(a,b) = 1 and k is not a multiple of three or i(a,b) bigger or equal to two and k equals plus or minus one. Further, we compute the index <X,Y> in <T_a,T_b> when <X,Y> is a proper subgroup.