Convex-compact subgroups of the Goeritz group

Tshishiku, Bena

doi:10.1090/tran/8905

Convex-compact subgroups of the Goeritz group
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by Bena Tshishiku;

Trans. Amer. Math. Soc. 377 (2024), 271-322

DOI: https://doi.org/10.1090/tran/8905

Published electronically: October 4, 2023

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Abstract:

Let $G<\operatorname {Mod}_2$ be the Goeritz subgroup of the genus-2 mapping class group. We show that finitely-generated, purely pseudo-Anosov subgroups of $G$ are convex cocompact in $\operatorname {Mod}_2$, addressing a case of a general question of Farb–Mosher. We also give a simple criterion to determine if a Goeritz mapping class is pseudo-Anosov, which we use to give very explicit convex-cocompact subgroups. In our analysis, a central role is played by the primitive disk complex $\mathcal {P}$. In particular, we (1) establish a version of the Masur–Minksy distance-formula for $\mathcal {P}$, (2) classify subsurfaces $X\subset S$ that are infinite-diameter holes of $\mathcal {P}$, and (3) show that $\mathcal {P}$ is quasi-isometric to a coned-off Cayley graph for $G$.

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