Orientable and non-orientable genus $n$ Wicks forms over hyperbolic groups

TL;DR

This paper extends the classical theory of Wicks forms, originally described for free groups, to hyperbolic groups, providing explicit forms and bounds for expressing elements as products of commutators and squares.

Contribution

It introduces a method to construct Wicks forms in hyperbolic groups and establishes bounds on conjugating element lengths for quadratic tuples.

Findings

Explicit lists of Wicks forms for a commutator and a square in hyperbolic groups

Bounds on lengths of conjugating elements for quadratic tuples

Extension of classical Wicks form theory to hyperbolic groups

Abstract

In 1962 M.J. Wicks gave a precise description of the form a commutator could take in a free group or a free product and in 1973 extended this description to cover a product of two squares. Subsequently, lists of "Wicks forms" were found for arbitrary products of commutators and squares in free groups and free products, by Culler, Vdovina and other authors. Here we construct Wicks forms for products of commutators and squares in a hyperbolic group. As applications we give explicit lists of forms for a commutator and for a square, and find bounds on the lengths of conjugating elements required to express a quadratic tuple of elements of a hyperbolic group as a Wicks form.