Genus n forms over Hyperbolic groups

TL;DR

This paper extends the classification of algebraic forms for commutators from free groups to hyperbolic groups, providing a comprehensive list of such forms in the latter context.

Contribution

It introduces a method for constructing forms of elements of any genus in hyperbolic groups, generalizing previous results from free groups and free products.

Findings

Constructed forms for elements of any genus in hyperbolic groups

Provided a complete list of commutator forms in hyperbolic groups

Extended classical results to a broader class of groups

Abstract

In 1962 M.J.Wicks gave a list of forms for commutators in both free groups and free products. Since then similar lists have been constructed for elements of higher genus. A. Vdovina has described a method for the construction of forms for elements of any genus in free products. We give a similar result for the construction of such forms in any hyperbolic group H and from this we obtain a full list of forms for commutators in H.