Effective construction of covers of canonical Hom-diagrams for equations over torsion-free hyperbolic groups

TL;DR

This paper presents an algorithm to construct covers of canonical solution diagrams for equations over torsion-free hyperbolic groups, enabling a comprehensive understanding of homomorphisms from finitely generated groups.

Contribution

It introduces a method to effectively build covers of canonical Hom-diagrams, enhancing the analysis of solutions over hyperbolic groups.

Findings

Algorithm constructs covers of canonical solution diagrams.

Encodes all homomorphisms as compositions through NTQ groups.

Provides a new characterization of $\Gamma$-limit groups.

Abstract

We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a cover of a canonical solution diagram. The diagram encodes all homomorphisms from $G$ to $\Gamma$ as compositions of factorizations through $\Gamma$-NTQ groups and canonical automorphisms of the corresponding NTQ-subgroups. We also give another characterization of $\Gamma$-limit groups as iterated generalized doubles over $\Gamma$.