Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

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+\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.