Cusped spaces for hierarchically hyperbolic groups, and applications to Dehn filling quotients

Abstract

We introduce a construction that simultaneously yields cusped spaces of relatively hyperbolic groups, and spaces quasi-isometric to Teichmueller metrics. We use this to study Dehn-filling-like quotients of various groups, among which mapping class groups of punctured spheres. In particular, we show that the mapping class group of a five-holed sphere (resp. the braid group on four strands) has infinite hyperbolic quotients (strongly) not isomorphic to hyperbolic quotients of any other given sphere mapping class group (resp. any other braid group). These quotients are obtained by modding out suitable large powers of Dehn twists, and we further argue that the corresponding quotients of the extended mapping class group have trivial outer automorphism groups. We obtain these results by studying torsion elements in the relevant quotients.

Figures (5)

• Figure 1: The blowup of two adjacent vertices of $\overline{X}$.
• Figure 2: Schematic representation of $\Pi$. Each cell describes how $\Pi\cap \operatorname{Cone}\left(v\right)$ is defined whenever $v\in\overline{X}^{(0)}$ belongs to the area given by the intersection between the row label and the column label: for example, if $v\in\overline{\Sigma}'\cap \overline{\Phi}$ we have that $\Pi\cap \operatorname{Cone}\left(v\right)=\Sigma'\cap \operatorname{Cone}\left(v\right)$.
• Figure 3: The various notions of hierarchical hyperbolicity at play in this paper, with arrows representing inclusions. Objects from the right column belong to the world of coarse geometry, while those in the middle and left columns are combinatorial in nature. At present, a robust definition of a relative combinatorial HHG has not been devised yet (hence we put this notion in a dashed box). Any reasonable notion should include both relative squid HHGs and combinatorial HHGs, and provide examples of relative HHGs.
• Figure 4: On the left, the situation in $\operatorname{Link}_{X}\left(\Pi\right)$. The geodesic $\gamma$ (the dashed line) only consists of $\mathcal{W}$-edges, because it lies in ${\operatorname{Link}_{\mathcal{C}(S)}\left(\overline{\Pi}\right)}^{+{\mathcal{W}}}$ and $\operatorname{Link}_{\mathcal{C}(S)}\left(\overline{\Pi}\right)$ is discrete. Furthermore, no point $w_i$ on $\gamma$ belongs to the saturation of $\Sigma$. On the right, the image of $\gamma$ under the quasi-isometry $\phi_\Pi$ is a quasi-geodesic $\gamma'$, which fellow-travels a geodesic $\eta$ (here, the continuous line). The bounded geodesic image theorem implies that all segments of $\gamma'$ outside a uniform neighbourhood of $\rho^{U_\Sigma}_{U_\Pi}$ (here, the dotted circle) have uniformly bounded projections to $\mathcal{C}(U_\Sigma)$. Furthermore, a uniformly bounded segment of $\gamma'$ can intersect said neighbourhood, and any two points in this segment have close projections to $\mathcal{C}(U_\Sigma)$ by how $\mathcal{W}$-edges are defined in the combinatorial structure.
• Figure 5: On the left, the half Dehn twists around the curves $\gamma_i$ generate the braid group. On the right, the pure braid group is generated by the Dehn twists around the curves $\eta^\pm_{i,j}$ for $1\le i<j\le n$ (here we only drew some). $\eta^+_{i,j}$ passes above the punctures between the $i$th and the $j$th, while $\eta^-_{i,j}$ passes below them.

Theorems & Definitions (133)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Remark 6
• Definition 1.1
• Definition 1.2: Coarse embedding
• Lemma 1.3: See proof of GrovesManning
• Theorem 1.4: GrovesManning