Metric equivalences of Heintze groups and applications to classifications in low dimension

2021/04/01 by Kivioja, Ville, Donne, Enrico Le, Golo, Sebastiano Nicolussi · 1 citation
#17B70 #20F67 #20F69 #22E25 #30L10 #53C23 #54E40 #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2104.00368

Abstract

We approach the quasi-isometric classification questions on Lie groups by considering low dimensional cases and isometries alongside quasi-isometries. First, we present some new results related to quasi-isometries between Heintze groups. Then we will see how these results together with the existing tools related to isometries can be applied to groups of dimension 4 and 5 in particular. Thus we take steps towards determining all the equivalence classes of groups up to isometry and quasi-isometry. We completely solve the classification up to isometry for simply connected solvable groups in dimension 4, and for the subclass of groups of polynomial growth in dimension 5.

Cited by

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

Related

• On quasi-isometry invariants associated to the derivation of a Heintze group
• Coarse differentiation and quasi-isometries of a class of solvable Lie groups I
• Quasi-isometries and rigidity of solvable groups
• Coarse differentiation and quasi-isometries of a class of solvable Lie groups II
• On the quasi-isometric and bi-Lipschitz classification of 3D Riemannian\n Lie groups