Dehn functions: computations, lower bounds, and the quasiisometric rigidity of \rmSol₅

2025/09/16 by Grayevsky, Ido, Pallier, Gabriel
#20F65 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2509.12823

Abstract

We establish distortion estimates in completely solvable Lie groups, using a sublinear bilipschitz retraction constructed by Cornulier, and interpolating between two theorems of Osin. This provides new lower bounds on Dehn functions. Our second main result is the quasiisometric rigidity of \rmSol₅ and its lattices. Together with a theorem of Peng, a key tool for the rigidity is the complete list of Dehn functions and dimensions of asymptotic cones of all simply connected solvable Lie groups of exponential growth up to dimension 5, which we compute using Cornulier and Tessera's results.

Citations

• Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups
• Metric equivalences of Heintze groups and applications to classifications in low dimension
• Aspects de la géométrie des groupes
• Assouad-Nagata dimension of connected Lie groups
• Coarse differentiation and quasi-isometries of a class of solvable Lie groups II
• Large scale Sobolev inequalities on metric measure spaces and applications
• Filling in solvable groups and in lattices in semisimple groups
• Strong rigidity ofQ-rank 1 lattices
• Isometry groups of Riemannian solvmanifolds

Related

• Dehn filling in semisimple Lie groups
• Generalized Dehn Functions II
• Scaled relators and Dehn functions for nilpotent groups
• The Dehn function of Sp(2n;Z)
• Filling functions of arithmetic groups