Giovanni Italiano

The Dehn function of subgroups of direct products of free groups

Subgroups of direct products of free groups can be very wild; however, under the hypothesis that they satisfy some finiteness conditions, they become much more controlled.

We are interested in studying the Dehn functions of such groups, which describes the geometry of their word problem. We prove uniform polynomial bounds for the Dehn function (under suitable finiteness hypothesis). More specifically, given n the number of free factors, if the subgroup is of type F_n-1, the Dehn function is bounded from above by N⁹ (our bound is actually stricter in most cases).

We also show an example of a subgroup whose Dehn function is exactly N⁴; the computation of the lower bound is based on an invariant built using braid groups.
This is joint work with Ascari, Bertolotti, Llosa-Isenrich and Migliorini.