Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures

TL;DR

This paper investigates the decidability of existential theories in torsion-free hyperbolic and relatively hyperbolic groups with abelian parabolics, and introduces an algorithm to identify groups hyperbolic relative to abelian subgroups.

Contribution

It proves the decidability of systems of equations in certain hyperbolic groups and presents a new algorithm for recognizing relative hyperbolic structures.

Findings

Decidability of satisfiability of equations in torsion-free hyperbolic groups.

Algorithm for detecting groups hyperbolic relative to abelian subgroups.

Improved understanding of the structure of relatively hyperbolic groups.

Abstract

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.