Tameness on the boundary and Ahlfors' measure conjecture

Tameness of hyperbolic 3-manifolds

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This answers a conjecture of Marden and implies the Ahlfors measure conjecture. Applications are given to other questions about Kleinian groups and 3-manifolds.

Marden's Tameness Conjecture: history and applications

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.

Drilling cores of hyperbolic 3-manifolds to prove tameness

We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ $δ$-hyperbolic and with Margulis constants independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding the core, we show that there exists an exiting sequence of surfaces $Σ_i$. We drill out the covers of $M_i$ by a core $C$ a…

The PL-methods for hyperbolic 3-manifolds to prove tameness

Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ negatively curved and with Margulis constant independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding to the core, we show that there exists an exiting sequence of surfaces $Σ_i$. Some of the ideas follow those of Agol. We drill out the c…

Existence of ruled wrappings in hyperbolic 3-manifolds

We present a short elementary proof of an existence theorem of certain CAT(-1)-surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem.

Anosov群的极限集上的二分法与测度

Dichotomy and measures on limit sets of Anosov groups

本文证明Anosov子群的共形测度支撑在极限集上当且仅当其维数临界，由此得到测度唯一性，并推广了Hopf-Tsuji-Sullivan二分法及Ahlfors测度猜想。