Orientable and non-orientable genus $n$ Wicks forms over 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.