Immersing almost geodesic surfaces in a closed hyperbolic three manifold

TL;DR

This paper constructs immersed surfaces in closed hyperbolic 3-manifolds that are almost geodesic, with injective fundamental group mappings, contributing to understanding surface embeddings in hyperbolic geometry.

Contribution

It introduces a method to construct immersed surfaces in hyperbolic 3-manifolds with injective fundamental group maps, advancing the study of surface embeddings.

Findings

Constructed closed surfaces immersed in hyperbolic 3-manifolds

Ensured the induced fundamental group maps are injective

Provided new techniques for surface immersion in hyperbolic geometry

Abstract

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.