Sublinear Bilipschitz Equivalence and the Quasiisometric Classification of Solvable Lie Groups (2410.05042v1)

Abstract: We prove a product theorem for sublinear bilipschitz equivalences which generalizes the classical work of Kapovich, Kleiner and Leeb on quasiisometries between product spaces. Independently we develop a new tool, based on a theorem of Cornulier on sublinear bilipschitz equivalences between solvable Lie groups, to evaluate the distortion of certain subgroups in central extensions. This is useful to provide lower bounds on Dehn functions. Building on work of Cornulier and Tessera, we compute the Dehn functions of all simply connected solvable Lie groups of exponential growth up to dimension $5$. We employ our product theorem to distinguish up to quasiisometry certain families among these groups which share the same dimension, cone-dimension and Dehn function. Finally, using a theorem of Peng together with our computations of cone-dimensions and Dehn functions, we establish the quasiisometric rigidity of the rank-two, five-dimensional solvable Lie group acting simply transitively by isometries on the horosphere in general position inside the product of three real hyperbolic planes.