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

Let $p$ be a point of an orientable hyperbolic $3$-manifold $M$, and let $m\ge 1$ and $k\ge 2$ be integers. Suppose that ${\alpha }_1,\dots ,{\alpha }_m$ 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 $[{\alpha }_1],\dots ,[{\alpha }_m]$, then $\bar{\chi }(G)\doteq -\chi (G)\le k-2$; 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 $\mathrm{\Delta }$ is a finite generating set for $G$, we define the $indexoffreedom$ $\mathrm{i}\mathrm{o}\mathrm{f}(\mathrm{\Delta })$ to be the largest integer $k$ such that $\mathrm{\Delta }$ contains $k$ elements that freely generate a rank-$k$ free subgroup of $G$. We define the $minimumindexoffreedom$ $\mathrm{m}\mathrm{i}\mathrm{o}\mathrm{f}(G)$ to be ${\min }_{\mathrm{\Delta }}\mathrm{i}\mathrm{o}\mathrm{f}(\mathrm{\Delta })$, where $\mathrm{\Delta }$ ranges over all finite generating sets for $G$. The result is that $\bar{\chi }(G)<\mathrm{i}\mathrm{o}\mathrm{f}(G)$. 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.