A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

Abstract

Abstract. We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level d mapping class group is the same as that of in the mapping class group.

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

“…Moreover, Rafi-Schleimer [18] proved that an orbifold covering map of orientable surfaces induces a quasi-isometric embedding between the mapping class groups. For examples of distorted subgroups of mapping class groups, see Broaddus-Farb-Putman [4], Cohen [5], and Kuno-Omori [12], where it is proved that the Torelli group « b g is distorted in Mod.S b g /. Moreover, it has been proved by that the handlebody group is exponentially distorted in the mapping class group of the boundary surface.…”

The mapping class group of a nonorientable surface is quasi-isometrically embedded in the mapping class group of the orientation double cover

“…For examples of distorted subgroups of mapping class groups, Broaddus-Farb-Putman [4], Cohen [5], and Omori and the second author [13] proved that the Torelli group I b g is distorted in Mod(S b g ). Moreover the handlebody group is exponentially distorted in the corresponding mapping class group by Hamenstädt-Hensel [12].…”