Hyperbolic surfaces and quadratic equations in groups

Abstract:

A group of a hyperbolic $2$-complex $K$ is a group with its associated van Kampen diagrams satisfying a hyperbolic curvature condition and a link condition on the degree of the interior vertices. A solution of an equation $({y_1}, \ldots ,{y_n}) = 1$ in $K$, where $W$ is a path in a $2$-complex $B$, is a mapping $\zeta :B \to K$ such that $\zeta W = W(\zeta {y_1}, \ldots ,\zeta {y_n})$ is contractible in $K$. This solution $\zeta$ is free if there is a mapping $h:B \to {K^{(1)}}$ such that $W(h{y_1}, \ldots ,h{y_n})$ is contractible in ${K^{(1)}}$ and such that $\zeta = \pi h$, where $\pi$ is the projection $\pi :{K^{(1)}} \to K$. Our main result is that each quadratic equation $W = 1$ has only finitely many nonfree solutions in $K$. Our tool is essentially the cancellation diagrams on surfaces developed by the present author based on work of Schupp.