Hyperbolic 3-manifolds with k-free fundamental group

Rosemary K Guzman

doi:10.1016/j.topol.2014.05.018

Abstract

We prove that if M is a closed, orientable, hyperbolic 3-manifold such that all subgroups of π1(M) of rank at most k=5 are free, then one can choose a point P in M such that the set of all elements of π1(M,P) that are represented by loops of length less than log9 is contained in a subgroup of rank at most 2; in particular, they generate a free group. In the 1990s, Culler, Shalen, and their co-authors initiated a program to understand the relationship between the topology and geometry of a closed hyperbolic 3-manifold; this paper extends those results to the setting of hyperbolic 3-manifolds with k=5-free fundamental group. A key ingredient in the proof is an analogue of a group-theoretic result of Kent and Louder–McReynolds concerning intersections and joins of rank three subgroups of a free group. Moreover, we state conjectural extensions of the 5-free result for values k>5, and establish them modulo the conjectured extension of the Kent and Louder–McReynolds result.

Highlights

The goal of this paper is to explore how the geometry of a closed, orientable hyperbolic 3manifold and its topological properties, especially its fundamental group, interact to provide new information about the manifold
One can express a hyperbolic n-manifold as the quotient of hyperbolic n-space modulo a discrete torsion-free group Γ of orientation-preserving isometries, in turn Γ is isomorphic to π1(M ); it is this vantage point that we take in this paper
We will say a group Γ is k-free, where k is a given positive integer, if every finitely generated subgroup of Γ of rank less than or equal to k is free. (Recall that the rank of a finitely generated group G is the minimal cardinality of a generating set for G.)

Summary

Introduction

The goal of this paper is to explore how the geometry of a closed, orientable hyperbolic 3manifold and its topological properties, especially its fundamental group, interact to provide new information about the manifold. If M is a closed, orientable, hyperbolic 3-manifold such that π1(M ) is k-free with k ≥ 5, when λ = log(2k − 1), there exists a point P in M such that the set of all elements of π1(M, P ) that are represented by loops of length less than λ is contained in a subgroup of π1(M ) of rank ≤ k − 3. A long range goal of the present work is to improve this bound with the added topological and geometric information that is gotten by virtue of the 5-free assumption and the rank ≤ 2 subgroup described in Theorem 1.4, with hopes that estimating the nearby and distant volumes of the given point P , under certain conditions, will lead to a refined lower bound on the global volume of M.

Lemma and Preliminaries

Γ-labeled complexes and Contractibility Arguments

Group-Theoretic Preliminaries

Findings

Theorem and general bound on rM