[Paper Review] Marden's Tameness Conjecture: history and applications

Richard D. Canary|arXiv (Cornell University)|Jul 31, 2010

Geometric and Algebraic TopologyMathematics90 references16 citations

TL;DR

This paper surveys the history and applications of Marden's Tameness Conjecture, which posits that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. The conjecture was recently proven by Agol and Calegari-Gabai, and the paper details its implications for the classification of hyperbolic 3-manifolds, spectral theory, group-theoretic properties, and deformation theory.

ABSTRACT

Marden's Tameness Conjecture predicts that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold. It was recently established by Agol and Calegari-Gabai. We will survey the history of work on this conjecture and discuss its many applications.

Motivation & Objective

• To trace the historical development of Marden's Tameness Conjecture and its role in the classification of hyperbolic 3-manifolds.
• To explain the significance of the conjecture in resolving long-standing problems in hyperbolic geometry and dynamics.
• To establish the equivalence between topological tameness and geometric tameness in hyperbolic 3-manifolds.
• To demonstrate how the Tameness Theorem enables a complete classification of hyperbolic 3-manifolds with finitely generated fundamental groups.
• To explore applications in deformation theory, including the Density Theorem and the structure of the space $AH(M)$

Proposed method

• Surveying foundational definitions and constructions in hyperbolic 3-manifold theory, including convex cores, conformal boundaries, and limit sets.
• Introducing the concept of topological tameness and contrasting it with geometric finiteness and convex cocompactness.
• Presenting the equivalence between geometric tameness (as defined by Thurston) and topological tameness, a key step in the proof.
• Applying the Tameness Theorem to prove Ahlfors' Measure Conjecture and to analyze the spectral theory of hyperbolic 3-manifolds.
• Using the Tameness Theorem to derive the Density Theorem for $AH(M)$, showing that the interior of $AH(M)$ is dense in its closure.
• Employing the Ending Lamination Theorem and convergence results of Thurston, Kleineidam-Souto, and Lecuire to construct limits of geometrically finite manifolds with prescribed end invariants.

Experimental results

Research questions

• RQ1 What is the relationship between topological tameness and geometric tameness in hyperbolic 3-manifolds?
• RQ2 How does the Tameness Theorem resolve Ahlfors' Measure Conjecture and other conjectures in hyperbolic geometry?
• RQ3 What is the structure of the deformation space $AH(M)$ for a compact hyperbolizable 3-manifold $M$?
• RQ4 How does the Tameness Theorem enable a complete classification of hyperbolic 3-manifolds with finitely generated fundamental group?
• RQ5 What are the group-theoretic consequences of tameness, particularly regarding separability and the finitely generated intersection property?

Key findings

• Marden’s Tameness Conjecture was proven by Agol and Calegari-Gabai, establishing that every hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact 3-manifold.
• The Tameness Theorem implies Ahlfors’ Measure Conjecture, which asserts that the limit set of a finitely generated Kleinian group has either measure zero or full measure on the Riemann sphere.
• The Tameness Theorem enables a complete classification of hyperbolic 3-manifolds with finitely generated fundamental group via the Ending Lamination Theorem.
• The Density Theorem for $AH(M)$ holds: the interior of $AH(M)$ is dense in its closure, and this follows from the Tameness Theorem and the Ending Lamination Theorem.
• When $ ho_1(M)$ is freely indecomposable, the components of $AH(M)$ are in one-to-one correspondence with $ar{rak{A}}(M)$, the quotient of the set of marked homeomorphism types under primitive shuffle equivalence.
• The space $AH(M)$ has infinitely many components if and only if $M$ has double trouble, i.e., there exist homotopic but non-boundary-homotopic simple closed curves on different boundary components, with one on a toroidal component.