# Distortion of surfaces in graph manifolds

## TL;DR

This paper investigates how the fundamental group of an immersed horizontal surface in a 3D graph manifold is distorted, showing quadratic distortion for virtually embedded surfaces and exponential distortion otherwise.

## Contribution

It establishes a clear distinction in the distortion types of surface groups based on their virtual embedding properties in graph manifolds.

## Key findings

- Quadratic distortion for virtually embedded surfaces
- Exponential distortion for non-virtually embedded surfaces
- Provides a classification of surface group distortions in graph manifolds

## Abstract

Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.