Dehn functions: computations, lower bounds, and the quasiisometric rigidity of ${\mathrm{S}\mathrm{o}\mathrm{l}}_5$

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 ${\mathrm{S}\mathrm{o}\mathrm{l}}_5$ 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.