Distortion of surfaces in 3-manifolds

TL;DR

This paper investigates how the fundamental group of a surface immersed in a non-geometric 3-manifold is distorted within the manifold's fundamental group, revealing the types of distortion possible and their relation to subgroup separability.

Contribution

It provides a classification of the possible distortion types for surface groups in non-geometric 3-manifolds and links distortion to subgroup separability properties.

Findings

Distortion types are linear, quadratic, exponential, or double exponential.

Distortion is closely related to the separability of the surface subgroup.

The paper characterizes the possible distortion behaviors in non-geometric 3-manifolds.

Abstract

Let $g \colon S \looparrowright N$ be a properly immersed $\pi_1$--injective surface in a non-geometric $3$--manifold $N$. We compute the distortion of $\pi_1(S)$ in $\pi_1(N)$ and show that how it is related to separability of $\pi_1(S)$ in $\pi_1(N)$. The only possibility of the distortion is linear, quadratic, exponential, and double exponential.