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

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

2011-09-23

arxiv.org/abs/1109.5158v1

Mathematics

Geometric Topology

Scientific paper

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 and are isomorphic. Moreover, we prove that = 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 in when is a proper subgroup.

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