Quadratic Equations in Hyperbolic Groups are NP-complete

Abstract

We prove that in a torsion-free hyperbolic group $\Gamma$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q{|}^3$ for an orientable equation, and by $N|Q{|}^4$ for a non-orientable equation, where $|Q|$ is the length of the equation, and the constant $N$ can be computed. We show that the problem, whether a quadratic equation in $\Gamma$ has a solution, is in NP, and that there is a PSpace algorithm for solving arbitrary equations in $\Gamma$. If additionally $\Gamma$ is non-cyclic, then this problem (of deciding existence of a solution) is NP-complete. We also give a slightly larger bound for minimal solutions of quadratic equations in a toral relatively hyperbolic group.

the paper will appear in the Transactions of the AMS, 2016