Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

Lewis Bowen

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Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

Lewis Bowen

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Abstract

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+ε) 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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Lewis Bowen. 2005-01-23. Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture. https://arxiv.org/abs/math/0411662

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