Algebraic groups over free and hyperbolic groups

TL;DR

This paper characterizes algebraic groups over torsion-free hyperbolic groups, providing a complete description when the variety is irreducible, thus advancing understanding of algebraic structures in geometric group theory.

Contribution

It introduces a formal definition of algebraic groups over arbitrary groups and fully describes these structures over torsion-free hyperbolic groups in the irreducible case.

Findings

Complete description of algebraic groups over torsion-free hyperbolic groups

Definition of algebraic groups as varieties with word map group laws

Characterization of irreducible varieties in this context

Abstract

We define an algebraic group over a group $G$ to be a variety - that is, a subset of $G^d$ defined by equations over $G$ - endowed with a group law whose coordinates can be expressed as word maps. In the case where $G$ is a torsion-free hyperbolic group and the underlying variety is irreducible we give a complete description of all algebraic groups.