Separability of embedded surfaces in 3-manifolds

TL;DR

This paper proves that properly embedded incompressible surfaces in compact 3-manifolds have fundamental groups that are separable within the manifold's fundamental group, advancing understanding of 3-manifold topology.

Contribution

It establishes the separability of the fundamental group of incompressible surfaces in 3-manifolds, a key property in 3-manifold topology.

Findings

Fundamental groups of incompressible surfaces are separable in the ambient manifold's fundamental group.

Supports the use of subgroup separability in 3-manifold topology.

Provides a foundation for further studies on subgroup separability in 3-manifolds.

Abstract

We prove that if S is a properly embedded incompressible surface in a compact 3-manifold M, then the fundamental group of S is separable in the fundamental group of M.