Structure of solutions of exponential equations in acylindrically hyperbolic groups

TL;DR

This paper analyzes the structure of solutions to exponential equations in acylindrically hyperbolic groups, providing a decomposition into independent equations and characterizing solution sets, especially in relatively hyperbolic groups.

Contribution

It introduces a decomposition of exponential equations into independent systems and characterizes their solution sets, extending results to relatively hyperbolic groups.

Findings

Solutions decompose into independent equations either loxodromic or elliptic.

Solution sets are $\

Solution sets are $\\mathbb{Z}$-semilinear in hyperbolic relative groups under certain conditions.

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.