Séance Séminaire

Séminaire Gaston Darboux

Dehn functions of central products of nilpotent groups

A longstanding open problem in geometric group theory is the quasi-isometry classification of finitely generated nilpotent groups. By Pansu's Theorem it was reduced to nilpotent groups with bilipschitz equivalent asymptotic cones. To make further progress on this problem, it is important to understand the quasi-isometry invariants that distinguish between them. One important quasi-isometry invariant are Dehn functions. They provide a quantitative measure for the difficulty of detecting if a word in its generators represents the trivial element of the group. In this talk, I will explain recent results that allow us to determine the Dehn functions of large classes of nilpotent groups arising as central products. As a consequence, for every $k>2$, we obtain many pairs of finitely presented $k$-nilpotent groups with bilipschitz asymptotic cones, but with different Dehn functions. This shows that Dehn functions can distinguish between nilpotent groups with the same asymptotic cone, making them interesting in the context of the conjectural quasi-isometry classification of nilpotent groups. This talk is based on joint works with García-Mejía, Pallier and Tessera.