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

TL;DR

This paper establishes bounds on the Euler characteristic of subgroups generated by short loops in hyperbolic 3-manifolds, linking geometric lengths to algebraic properties, and relates these to Wilson's Freiheitssatz.

Contribution

It introduces new bounds connecting loop lengths in hyperbolic 3-manifolds with subgroup Euler characteristics, extending Wilson's Freiheitssatz to geometric contexts.

Findings

Bound on subgroup Euler characteristic based on loop length

Introduction of index of freedom and minimum index of freedom concepts

Equivalence to a special case of 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.