Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

Details Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

2004-11-30

arxiv.org/abs/math/0411662v5

Mathematics

Geometric Topology

54 pages

Scientific paper

We prove the following: 1. Let epsilon>0 and let S_1,S_2 be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'_i of S_i (for i=1,2) such that there is a (1+\epsilon) bi-Lipschitz homeomorphism between S'_1 and S'_2 and both covers S'_i have bounded injectivity radius. 2. Let M be a closed hyperbolic 3-manifold. Then there exists a map j: S -> M where S is a surface of bounded injectivity radius and j is a pi_1-injective local isometry onto its image.

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