We show that the Dehn function of the handlebody group is exponential in any genus . On the other hand, we show that the handlebody group of genus 2 is cubical and biautomatic and therefore has a quadratic Dehn function.
"The Geometry of the Handlebody Groups II: Dehn Functions." Michigan Math. J. 70 (1) 23 - 53, March 2021. https://doi.org/10.1307/mmj/1586160015