Enric Ventura1
In this talk we will consider
Dehn functions of groups. After a quick review of their interest and main
properties, we'll introduce the much more modern notion of "mean Dehn
function". In the second part of the talk, we'll scketch the proof of the
following theorem: "the mean Dehn function of a finitely generated abelian
group is )". This result makes a big contrast with the well known
facts that the Dehn function of abelian groups is quadratic, and that there
is no group with Dehn function between quadratic and linear. The proof
consists on a detailed combinatorial analysis involving several countings of
paths of given length in the integral lattice
.