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arXiv · 2510.12161

Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth

Katrin FÄssler·Enrico Le Donne·Sebastiano Nicolussi Golo·Alessandro Ottazzi·Pierre Pansu
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Abstract

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.

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Katrin FÄssler, Enrico Le Donne, Sebastiano Nicolussi Golo, Alessandro Ottazzi, Pierre Pansu. 2025-10-14. Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth. https://arxiv.org/abs/2510.12161

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