Bounded combinatorics and uniform models for hyperbolic 3-manifolds

Abstract

Bounded-type 3-manifolds arise as combinatorially bounded gluings of irreducible 3-manifolds chosen from a finite list. We prove effective hyperbolization and effective rigidity for a broad class of 3-manifolds of bounded type and large gluing heights. Specifically, we show the existence and uniqueness of hyperbolic metrics on 3-manifolds of bounded type and large heights, and prove existence of a bilipschitz diffeomorphism to a combinatorial model described explicitly in terms of the list of irreducible manifolds, the topology of the identification, and the combinatorics of the gluing maps.

Subgroups of mapping class groups related to Heegaard splittings and bridge decompositions

“…its generalisation by Namazi and Souto [30] and Brock et al [4]. We believe that we can obtain the same result for bridge decompositions since their theory is valid in more general settings including the case where the hyperbolic manifolds have torus cusps as is suggested in [4].…”

“…4 We shall first show that there are A, B as above such that c ∪ d ∪ f is an (A, B)-quasi-geodesic. By Lemma 1.1, except for geodesic segments starting from x of length at most L , one on c and the other on d, each point of c ∪ d is within distance δ from e. In the same way, we see that by Lemma 1.2, except for geodesic segments starting from y of length L , one on e and the other on f , every point on e ∪ f is within distance δ from h. These bounds imply that…”

“…We also note that Theorems C and E below may be regarded as a variant of the asymptotical faithfulness of the homomorphism π 1 (H i ) → π 1 (M) established by Namazi The authors would like to thank Brian Bowditch for his essential contribution to the proof of Theorem B, without which they should not have been able to complete the work. They would also like to thank Jeff Brock and Yair Minsky for stimulating conversation, valuable comments on the first version of this paper, and allowing them to read a draft of their joint work [4] with Hossein Namazi and Juan Souto on model manifolds, on which Theorems C and D depend. Finally, they would like to express their gratitude to the referees for their careful reading of the manuscript and very helpful suggestions.…”

Margulis spacetimes via the arc complex

“…Lemma A. 4 The vector w 0 ∈ R 2,1 is timelike and satisfies H 2 ∩ Rw 0 = {p}, where p ∈ H 2 is the intersection point of the common perpendicular toβ 1 and β 3 in H 2 with the common perpendicular toβ 2 andβ 4 .…”

“…While coarse models, like the curve complex or pants complex, are used to great effect in the study of the various metrics and compactifications of Teichmüller spaces (see [3,4,36,42] for instance), parameterizations and cellulations can provide insight at both macroscopic and microscopic scales. Prominent examples are the two (mutually homeomorphic) cell decompositions of the decorated Teichmüller space of a punctured surface described by Harer [29] and Penner [39] (see also [28,30] for generalizations), which have interesting applications to mapping class groups (see [40]).…”