A Combination Theorem for Geodesic Coarsely Convex Group Pairs

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.