Resolutions of orbifold generalised triangle groups

Abstract

Let $T=\langle f,g\mid f^{l}=g^{m}=W^{n}=1\rangle$ be a generalised triangle group in which $W$ is the Farey word of a rational slope $q=r/s$, and suppose that $T$ is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let $G=\langle f,g\mid f^{l}=g^{m}=W^{n}\rangle$ be the associated central extension. We compute an explicit free resolution of the trivial module over $\mathbb{Z}G$ of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over $\mathbb{Z}T$ which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family $q=0/1$ in 1983. Since then only the family $q=1/2$ has been treated, by Ellis and Williams, and for that family our resolution over $\mathbb{Z}T$ has strictly smaller rank in every degree.