Euler characteristics, lengths of loops in hyperbolic 3-manifolds, and Wilson's Freiheitssatz

Abstract

Let $p$ be a point of an orientable hyperbolic $3$-manifold $M$, and let $m\ge1$ and $k\ge2$ be integers. Suppose that $\alpha_1,\ldots,\alpha_m$ are loops based at $p$ having length less than $\log(2k-1)$. We show that if $G$ denotes the subgroup of $\pi_1(M,p)$ generated by $[\alpha_1],\ldots,[\alpha_m]$, then $\overline{\chi}(G)\doteq-\chi(G)\le k-2$; here $\chi(G)$ denotes 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 $\Delta$ is a finite generating set for $G$, we define the $index\ of\ freedom$ ${\rm iof}(\Delta)$ to be the largest integer $k$ such that $\Delta$ contains $k$ elements that freely generate a rank-$k$ free subgroup of $G$. We define the $minimum\ index\ of\ freedom$ ${\rm miof}(G)$ to be $\min_{\Delta }{\rm iof}(\Delta )$, where $\Delta $ ranges over all finite generating sets for $G$. The result is that $\overline{\chi}(G)<{\rm iof}(G)$. The author has recently learned that this is equivalent to a special case of a theorem about arbitrary finitely presented groups due to J. S. Wilson.