Isoperimetric inequalities for the handlebody groups (Q2847196)

The handlebody group is the group of isotopy classes of orientation preserving homeomorphisms of a handlebody. This group can be identified with a sub-group of the mapping class group of the boundary of the handlebody. However, the large scale properties of these two groups are very different. Indeed it is known that in genus at least three, the handlebody group is exponentially distorted in the mapping class group of the boundary.NEWLINENEWLINEThe mapping class group has quadratic Dehn function. In the present paper it is proved that the handlebody group has at most exponential Dehn function. In other words, the handlebody group satisfies an exponential isoperimetric inequality. It should be noted that for rank at least three, the group of outer automorphisms of a free group has exponential Dehn function, but since the kernel of the projection from the handlebody group to the outer automorphims group is infinite, this gives no restrictions on the Dehn function of the handlebody group.