Euler characteristics, free subgroups, and lengths of loops in hyperbolic 3-manifolds

Abstract

Let p be a point of an orientable hyperbolic 3-manifold M , and let m ≥ 1 and k ≥ 2 be integers. Suppose that α 1 , . . . , α m are loops based at p having length less than log(2k −1). We show that if G denotes the subgroup of πdenotes the Euler characteristic of the group G, which is always defined in this situation.This result is deduced from a result about an arbitrary finitely generated subgroup G of the fundamental group of an orientable hyperbolic 3-manifold. If ∆ is a finite generating set for G, we define the index of freedom iof(∆) to be the largest integer k such that ∆ contains k elements that freely generate a rank-k free subgroup of G. We define the minimum index of freedom miof(G) to be min ∆ iof(∆), where ∆ ranges over all finite generating sets for G. The result is that χ(G) < miof(G).