Proper Affine Deformations of the One-Holed Torus

Abstract

A Margulis spacetime is a complete flat Lorentzian 3-manifold M with free fundamental group. Associated to M is a noncompact complete hyperbolic surface Σ homotopy-equivalent to M . The purpose of this paper is to classify Margulis spacetimes when Σ is homeomorphic to a one-holed torus. We show that every such M decomposes into polyhedra bounded by crooked planes, corresponding to an ideal triangulation of Σ. This paper classifies and analyzes the structure of crooked ideal triangles, which play the same role for Margulis spacetimes as ideal triangles play for hyperbolic surfaces. This extends our previous work on affine deformations of three-holed sphere and two-holed cross surfaces. arXiv:1501.04535v1 [math.DG] 19 Jan 2015 45 10.2. Geometric tameness 45 References 47 2. Notation and terminology 2.1. Elementary conventions. By a triple (respectively pair ) we shall always mean an ordered triple or pair respectively, unless otherwise stated.

Topological tameness of Margulis Spacetimes

“…With Theorem 1.1, we can obtain compactifications, i.e., Theorem 1.2 also. For rank two cases, the conjecture was verified by Charette-Drumm-Goldman [17]. Related to this conjecture, we have Proposition 7.3.…”

Lengthening deformations of singular hyperbolic tori

“…(A) to prove that for every hyperbolic metric g on S, the polygon Π has one side associated to each simple closed curve in S (this recovers one of the main results of [CDG3], and also follows from a special case of [DGK2]: see §6 therein); (B) to extend this result to the case when the boundary component of S is replaced with a cone singularity; (C) to provide formulas and estimates for quantities such as the side lengths of the infinite polygon Π (in appropriate affine charts); (D) to analyze how Π degenerates when the area of the singular hyperbolic surface S goes to 0.…”

“…As early as 2003, Charette [C] proved that the simple closed curves corresponding to the sides of Π can have arbitrarily long expressions in the generators of π 1 (S). Charette, Drumm and Goldman prove in [CDG3] that in fact, all simple closed curves arise. Our purpose in this note is:…”

Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane

“…Alternatively, a superbasis is an Inn e (F 2 )-equivalence class of basic triples, where Inn e (F 2 ) is defined in (5) and e is the elliptic involution defined in (4). That is, two basic triples (X, Y, Z) and (X , Y , Z ) represent the same superbasis if and only if (8) [4].) Geometrically, a superbasis corresponds to an ordered triple of isotopy classes of unoriented simple loops on Σ 1,1 with mutual geodesic intersection numbers 1.…”