Curvature distribution, relative presentations and hyperbolicity with an application to Fibonacci groups

Abstract

Given a finite presentation for a group G which can also be realised as a relative presentation we give conditions on the relative presentation which, if satisfied, proves G hyperbolic. Using a curvature distribution method we confirm these conditions for the length four one-relator relative presentation 〈u,t|tn,tmutu−r〉 for many values of r and n deduce that the corresponding generalised Fibonacci groups F(r,n) are hyperbolic.