Margulis numbers for Haken manifolds

Marc Culler,Peter B Shalen

doi:10.1007/s11856-011-0189-z

Abstract

For every closed orientable hyperbolic Haken 3-manifold and, more generally, for any orientable hyperbolic 3-manifold M which is homeomorphic to the interior of a Haken manifold, the number 0.286 is a Margulis number. If H 1(M;ℚ) ≠ 0, or if M is closed and contains a semi-fiber, then 0.292 is a Margulis number for M.

Highlights

If M is an orientable hyperbolic n-manifold, we may write M = Hn/Γ where Γ ≤ Isom+(Hn) is discrete and torsion-free
To prove Theorem 1.1 for a hyperbolic manifold M = H3/Γ which is homeomorphic to the interior of a given Haken manifold N, we must obtain a lower bound for max(dP (x), dP (y)) whenever P is a point of H3 and x and y are non-commuting elements of Γ
The decomposition Γ = X ∪Y ∪C gives rise to a decomposition of the area measure on S∞, and the set-theoretic conditions (i)—(iii) give information about how the terms in the decomposition transform under x and y. This information is used to obtain a lower bound for max(dP (x), dP (y)); here the hyperbolic trigonometry is combined with an argument analogous to the one used in [8]

Summary

Introduction

To prove Theorem 1.1 for a hyperbolic manifold M = H3/Γ which is homeomorphic to the interior of a given Haken manifold (or strict Haken manifold) N , we must obtain a lower bound for max(dP (x), dP (y)) whenever P is a point of H3 and x and y are non-commuting elements of Γ. The decomposition Γ = X ∪Y ∪C gives rise to a decomposition of the area measure on S∞, and the set-theoretic conditions (i)—(iii) give information about how the terms in the decomposition transform under x and y This information is used to obtain a lower bound for max(dP (x), dP (y)); here the hyperbolic trigonometry is combined with an argument analogous to the one used in [8]. We thank the anonymous referee for several helpful suggestions which improved the exposition

Measures and displacements

Group actions on trees

Incompressible surfaces and actions on trees

Word growth and displacements

Decompositions

Some hyperbolic trigonometry

Proof of the main theorem