Structure of solutions of exponential equations in acylindrically hyperbolic groups

Abstract

Let $G$ be a group acting acylindrically on a hyperbolic space and let $E$ be an exponential equation over $G$. We show that $E$ is equivalent to a finite disjunction of finite systems of pairwise independent equations which are either loxodromic over virtually cyclic subgroups or elliptic. We also obtain a description of the solution set of $E$. We obtain stronger results in the case where $G$ is hyperbolic relative to a collection of peripheral subgroups $\{H_{\lambda }{\}}_{\lambda \in \Lambda }$. In particular, we prove in this case that the solution sets of exponential equations over $G$ are $\mathbb{Z}$-semilinear if and only if the solution sets of exponential equations over every $H_{\lambda }$, $\lambda \in \Lambda$, are $\mathbb{Z}$-semilinear. We obtain an analogous result for finite disjunctions of finite systems of exponential equations and inequations over relatively hyperbolic groups in terms of definable sets in the weak Presburger arithmetic.

18 pages, 2 figures