A Combination Theorem for Geodesic Coarsely Convex Group Pairs

TL;DR

The paper develops a relative geometric framework for group pairs $(G,oldsymbol{\mathcal{H}})$, introducing the notions of weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex group pairs. It proves a combination theorem for graphs of groups: if each vertex pair $(G_v,oldsymbol{\mathcal{H}}_v)$ lies in one of these classes and every edge group embeds into a peripheral member, then the whole pair $(G,oldsymbol{\mathcal{H}})$ lies in the same class with a finite peripheral structure $oldsymbol{\mathcal{H}}$, with a compatibility condition on conjugacy to peripheral subgroups. The construction relies on equivariant trees of spaces obtained via pushouts, coned-off geometries, and Spike/Coalescence techniques for both amalgamated products and HNN-extensions, and shows how to assemble vertex-space bicombings into a global bicombing on the tree of spaces, preserving quasi-geodesicity, consistency, and geodesic coarse convexity. These results extend Alonso–Bridson and Fukaya–Matsuka style combination theorems to coarsely convex group pairs and prepare the ground for relatives of the coarse Baum–Connes conjecture in this setting. The framework provides tools for building new examples of coarsely convex group pairs and for studying their boundaries and relative geometric actions.

Abstract

The first author and Oguni introduced a class of groups of non-positive curvature, called coarsely convex group. The recent success of the theory of groups which are hyperbolic relative to a collection of subgroups has motivated the study of other properties of groups from the relative perspective. In this article, we propose definitions for the notions of weakly semihyperbolic, semihyperbolic, and coarsely convex group pairs extending the corresponding notions in the non-relative case. The main result of this article is the following combination theorem. Let $\mathcal{A}_{wsh}$, $\mathcal{A}_{sh}$, and $\mathcal{A}_{gcc}$ denote the classes of group pairs that are weakly semihyperbolic, semihyperbolic, and geodesic coarsely convex respectively. Let $\mathcal A$ be one of the classes $\mathcal{A}_{wsh}$, $\mathcal{A}_{sh}$, and $\mathcal{A}_{gcc}$. Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a non-trivial finite collection of subgroups $\mathcal{H}$ of $G$ satisfying the following properties. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal{A} $, then $(G,\mathcal{H})$ is in $\mathcal{A}$. The main results of the article are combination theorems generalizing results of Alonso and Bridson; and Fukaya and Matsuka.

Figures (7)

• Figure 1: This figure illustrates the structure of the Spike space. Let $x,y,z \in X^H$. From $x$, there emanete $G_x/C$ edge, each terminating at a spike-point. The stabilizer of each spike-point is of the form $gC$ for some $g \in G_x$.
• Figure 2: Proof of Proposition \ref{['cft']}-Case \ref{['ExitemI']})
• Figure 3: Proof of Proposition \ref{['cft']}-Case \ref{['ExitemII']})
• Figure 4: Proof of Proposition \ref{['cft']}-Case \ref{['ExitemIII']})
• Figure 5: Proof of Proposition \ref{['cbt']}-Case \ref{['EyitemI']}

Theorems & Definitions (20)

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• proof : Proof of Proposition \ref{['cft']}
• proof : Proof of Proposition \ref{['cbt']}