Combining relatively hyperbolic groups over a complex of groups

TL;DR

The paper proves a relative hyperbolicity result for the fundamental group $G=\pi_1(G(\mathcal{Y}))$ of a nonpositively curved developable complex of groups, under the assumptions that each vertex group is relatively hyperbolic and edge maps are inclusions of full relatively quasiconvex subgroups, with the universal cover $X$ being δ-hyperbolic and the $G$-action acylindrical. The authors extend prior work by constructing a Bowditch boundary model $\overline{Z}$ via gluing Bowditch boundaries of simplex stabilizers, and show that $G$ acts on $\overline{Z}$ as a geometrically finite convergence group; applying Yaman then yields that $G$ is relatively hyperbolic. The construction also yields a structural decomposition of Bowditch boundaries in terms of the cell complex and stabilizer boundaries, offering a topological lens for understanding boundaries in complex-of-groups contexts. Overall, the work provides a dynamical, boundary-based combination theorem for relatively hyperbolic groups along with a general mechanism for boundary decomposition in cell complexes.

Abstract

Given a complex of groups $G(\mathcal{Y}) = (G_σ, ψ_a, g_{a,b})$ where all $G_σ$ are relatively hyperbolic, the $ψ_a$ are inclusions of full relatively quasiconvex subgroups, and the universal cover $X$ is CAT$(0)$ and $δ$--hyperbolic, we show $π_1(G(\mathcal{Y}))$ is relatively hyperbolic. The proof extends the work of Dahmani and Martin by constructing a model for the Bowditch boundary of $π_1(G(\mathcal{Y}))$. We prove the model is a compact metrizable space on which $G$ acts as a geometrically finite convergence group, and a theorem of Yaman then implies the result. More generally, this model shows how any suitable action of a relatively hyperbolic group on a simply connected cell complex encodes a decomposition of the Bowditch boundary into the boundary of the cell complex and the boundaries of cell stabilizers. We hope this decomposition will be helpful in answering topological questions about Bowditch boundaries.

Figures (9)

• Figure 1: On the left is $\mathbb{H}^2$ tiled by octagons, describing an action of $G$. The red curves represent lifts of $c$. In the center is the tree of circles we get by contracting the endpoints of each lift of $c$. These images come from benzvi2022hyperbolicboundariesvshyperbolic. On the right is part of the Bass--Serre tree of the splitting for $G$. Red and blue vertices are stabilized by conjugates of $[a_1,b_1]$ and $[a_2,b_2]$ respectively.
• Figure 2: The situation of \ref{['Short Paths of Simplices']}.
• Figure 3: From left to right, a simplex, the scwol constructed by reverse inclusion, and the geometric realization of this scwol.
• Figure 4: On the left is a geometric simplex of $X$. On the right, each circle represents the Bowditch boundary of the corresponding cell. The arrows represent the maps $\varphi_{\tau,\tau'}$ between boundaries, and $\partial_{Stab}G$ is constructed by identifying points along these arrows. construct $Z$.
• Figure 5: In this example, $D(\xi)$ is the single bold edge, $N(\xi)$ is the bold edge together with the interior of each of the thinner edges, and $Lk(\xi)$ is only the interior of these thinner edges. The neighborhood $D^\varepsilon(\xi)$ is everything contained in the dotted region.

Theorems & Definitions (268)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Convergence Groups
• Definition 2.2: Conical and Parabolic Limit Points
• Definition 2.3: Geometrically Finite
• Definition 2.4: Relatively Hyperbolic
• Theorem 2.5
• Proposition 2.6
• Definition 2.7
• Definition 2.8