Dehn Functions: Computations, Lower Bounds, and the Quasiisometric Rigidity of SOL 5

Cornulier and Tessera established that the Dehn function of a solvable Lie group is either exponential or polynomially bounded, and gave criteria to decide between these two outcomes. In this paper, we make their result more explicit regarding the polynomial order of the Dehn function, and improve their lower bound in some cases. Along the way, we devise a practical method to compute distortion estimates in solvable Lie groups, of independent interest. Using Cornulier and Tessera’s methods we give exact estimates on the Dehn functions of solvable Lie groups of dimension at most [Formula: see text], and as an application, we prove the QI rigidity of the group [Formula: see text] and its lattices. The latter QI rigidity result also uses works of Peng.