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

TL;DR

This paper investigates subgroups of the mapping class group generated by specific Dehn twists and their powers, establishing conditions for isomorphism and computing subgroup indices in various cases.

Contribution

It provides new results on the structure and index of subgroups generated by powers of Dehn twists in the mapping class group, including isomorphism conditions.

Findings

<X,Y> is isomorphic to <T_a,T_b> in many cases.

<X,Y> equals <T_a,T_b> when intersection number i(a,b)=1 and k not divisible by 3.

Computed the index of <X,Y> in <T_a,T_b> when <X,Y> is a proper subgroup.

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.