Quasiconformal homogeneity and subgroups of the mapping class group

Abstract

In the vein of Bonfert-Taylor, Bridgeman, Canary, and Taylor we introduce the notion of quasiconformal homogeneity for closed oriented hyperbolic surfaces restricted to subgroups of the mapping class group. We find uniform lower bounds for the associated quasiconformal homogeneity constants across all closed hyperbolic surfaces in several cases, including the Torelli group, congruence subgroups, and pure cyclic subgroups. Further, we introduce a counting argument providing a possible path to exploring a uniform lower bound for the nonrestricted quasiconformal homogeneity constant across all closed hyperbolic surfaces.

Figures (2)

• Figure 1: A 4-punctured sphere in $X$ with $\gamma$ bounding two embedded pairs of pants. The curve $\alpha$ intersects $\gamma$ once and spirals towards both $\beta_1$ and $\beta_2$ so that it is disjoint from all boundary components.
• Figure 2: Lifts of $\alpha$ and $\gamma$ in the upper half plane. Also drawn is a copy of $\tilde{\alpha}$ under a translation by the element of $\pi_1X$ representing $\gamma$. The dotted geodesic is the image of $\tilde{\alpha}$ under the lift of a Dehn twist about $\gamma$.

Theorems & Definitions (36)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3: Theorem 1.1 in bcmt
• Proposition 2.4: Proposition 3.2 in bbc
• Lemma 3.1: Wolpert's Lemma, Lemma 12.5 in fm