The Geometry of the Handlebody Groups II: Dehn functions

TL;DR

This paper investigates the Dehn functions of handlebody groups, demonstrating exponential growth for genus ≥ 3 and quadratic growth for genus 2, revealing geometric and algebraic properties of these groups.

Contribution

It establishes the exponential Dehn function for handlebody groups of genus ≥ 3 and proves that genus 2 handlebody groups are cubical and biautomatic with quadratic Dehn functions.

Findings

Handlebody groups of genus ≥ 3 have exponential Dehn functions.

Genus 2 handlebody groups are cubical and biautomatic.

Genus 2 handlebody groups have quadratic Dehn functions.

Abstract

We show that the Dehn function of the handlebody group is exponential in any genus $g\geq 3$. On the other hand, we show that the handlebody group of genus $2$ is cubical, biautomatic, and therefore has a quadratic Dehn function.