The geometry of the handlebody groups I: Distortion

TL;DR

This paper proves that the handlebody groups are exponentially distorted within the mapping class groups of their boundary surfaces, revealing significant geometric complexity differences.

Contribution

It establishes the exponential distortion of handlebody groups in boundary surface mapping class groups, extending understanding of their geometric properties.

Findings

Handlebody groups are exponentially distorted in boundary surface mapping class groups.

The result applies to handlebodies of genus at least 2 with marked points.

Solid tori with at least two marked points also exhibit this distortion.

Abstract

We show that the mapping class group of a handlebody of genus at least 2 (with any number of marked points or spots) is exponentially distorted in the mapping class group of its boundary surface. The same holds true for solid tori with at least two marked points or spots.