Weak Forms of the Ehrenpreis Conjecture and the Surface Subgroup Conjecture

الأصل · EN

We prove the following: 1. Let epsilon>0 and let S₁,S₂ be two closed hyperbolic surfaces. Then there exists locally-isometric covers S'ᵢ of Sᵢ (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 pi₁-injective local isometry onto its image.