2025/10/14 by Katrin Fässler, FÄssler, Katrin, Enrico Le Donne +7
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Equations Stability Results #Group Theory (math.GR) #Meromorphic and Entire Functions #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2510.12161
openalex publication_date 2025/10/14 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28
We study conditions under which quasi-conformal homeomorphisms are quasi-isometries. We show that if two nilpotent geodesic Lie groups are quasi-conformally homeomorphic, then they are quasi-isometrically equivalent. We also give more general results beyond the nilpotent case. In particular, we show that quasi-conformal homeomorphisms between geodesic Lie groups are quasi-isometries whenever the spaces have strict parabolic or hyperbolic conformal type. As a consequence, quasi-conformal homeomorphisms between geodesic Lie groups with infinite fundamental group are quasi-isometries. The statements for Lie groups are deduced from a more general study on metric measure spaces with uniformly locally bounded geometry.