Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909844145963008 |
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| author | FÄssler, Katrin Donne, Enrico Le Golo, Sebastiano Nicolussi Ottazzi, Alessandro Pansu, Pierre |
| author_facet | FÄssler, Katrin Donne, Enrico Le Golo, Sebastiano Nicolussi Ottazzi, Alessandro Pansu, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth FÄssler, Katrin Donne, Enrico Le Golo, Sebastiano Nicolussi Ottazzi, Alessandro Pansu, Pierre Metric Geometry Group Theory 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. |
| title | Quasi-conformal VS quasi-isometric equivalence in spaces with controlled growth |
| topic | Metric Geometry Group Theory |
| url | https://arxiv.org/abs/2510.12161 |