Foliations for solving equations in groups: free, virtually free, and hyperbolic groups

TL;DR

This paper introduces algorithms for solving equations and inequations with rational constraints in virtually free and hyperbolic groups, utilizing Rips classification and canonical representatives to extend solutions to complex group structures.

Contribution

It presents a novel algorithmic approach based on Rips classification for solving equations in virtually free and hyperbolic groups, including torsion cases and rational constraints.

Findings

Algorithm successfully solves equations in virtually free groups.

Extension to hyperbolic groups with torsion achieved.

Handles quasi-isometrically embeddable rational constraints.

Abstract

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.