From homogeneous metric spaces to Lie groups
We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively.After a review of a number of classical results, we use the Gleason–Iwasawa–Montgomery–Yamabe–Zippin structure theory to show that for all positive $ \epsilon $, each...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.608/ |
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| author | Cowling, Michael G. Kivioja, Ville Le Donne, Enrico Nicolussi Golo, Sebastiano Ottazzi, Alessandro |
| author_facet | Cowling, Michael G. Kivioja, Ville Le Donne, Enrico Nicolussi Golo, Sebastiano Ottazzi, Alessandro |
| author_sort | Cowling, Michael G. |
| collection | DOAJ |
| description | We study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively.After a review of a number of classical results, we use the Gleason–Iwasawa–Montgomery–Yamabe–Zippin structure theory to show that for all positive $ \epsilon $, each such space is $ (1,\epsilon ) $-quasi-isometric to a connected metric Lie group (metrized with a left-invariant distance that is not necessarily Riemannian).Next, we develop the structure theory of Lie groups to show that every homogeneous metric manifold is homeomorphically roughly isometric to a quotient space of a connected amenable Lie group, and roughly isometric to a simply connected solvable metric Lie group.Third, we investigate solvable metric Lie groups in more detail, and expound on and extend work of Gordon and Wilson [31, 32] and Jablonski [44] on these, showing, for instance, that connected solvable Lie groups may be made isometric if and only if they have the same real-shadow.Finally, we show that homogeneous metric spaces that admit a metric dilation are all metric Lie groups with an automorphic dilation. |
| format | Article |
| id | doaj-art-52aa1fd9784944c396dc3bf16806ee7f |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-52aa1fd9784944c396dc3bf16806ee7f2025-02-07T11:23:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G9943101410.5802/crmath.60810.5802/crmath.608From homogeneous metric spaces to Lie groupsCowling, Michael G.0Kivioja, Ville1Le Donne, Enrico2Nicolussi Golo, Sebastiano3Ottazzi, Alessandro4School of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, AustraliaDepartment of Mathematics and Statistics, University of Jyväskylä, Jyväskylä FI-40014 FinlandDepartment of Mathematics and Statistics, University of Jyväskylä, Jyväskylä FI-40014 Finland; Department of Mathematics, University of Fribourg, Fribourg CH-1700 SwitzerlandDepartment of Mathematics and Statistics, University of Jyväskylä, Jyväskylä FI-40014 FinlandSchool of Mathematics and Statistics, University of New South Wales, UNSW Sydney 2052, AustraliaWe study homogeneous metric spaces, by which we mean connected, locally compact metric spaces whose isometry group acts transitively.After a review of a number of classical results, we use the Gleason–Iwasawa–Montgomery–Yamabe–Zippin structure theory to show that for all positive $ \epsilon $, each such space is $ (1,\epsilon ) $-quasi-isometric to a connected metric Lie group (metrized with a left-invariant distance that is not necessarily Riemannian).Next, we develop the structure theory of Lie groups to show that every homogeneous metric manifold is homeomorphically roughly isometric to a quotient space of a connected amenable Lie group, and roughly isometric to a simply connected solvable metric Lie group.Third, we investigate solvable metric Lie groups in more detail, and expound on and extend work of Gordon and Wilson [31, 32] and Jablonski [44] on these, showing, for instance, that connected solvable Lie groups may be made isometric if and only if they have the same real-shadow.Finally, we show that homogeneous metric spaces that admit a metric dilation are all metric Lie groups with an automorphic dilation.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.608/Homogeneous spacesStructureLie groups |
| spellingShingle | Cowling, Michael G. Kivioja, Ville Le Donne, Enrico Nicolussi Golo, Sebastiano Ottazzi, Alessandro From homogeneous metric spaces to Lie groups Comptes Rendus. Mathématique Homogeneous spaces Structure Lie groups |
| title | From homogeneous metric spaces to Lie groups |
| title_full | From homogeneous metric spaces to Lie groups |
| title_fullStr | From homogeneous metric spaces to Lie groups |
| title_full_unstemmed | From homogeneous metric spaces to Lie groups |
| title_short | From homogeneous metric spaces to Lie groups |
| title_sort | from homogeneous metric spaces to lie groups |
| topic | Homogeneous spaces Structure Lie groups |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.608/ |
| work_keys_str_mv | AT cowlingmichaelg fromhomogeneousmetricspacestoliegroups AT kiviojaville fromhomogeneousmetricspacestoliegroups AT ledonneenrico fromhomogeneousmetricspacestoliegroups AT nicolussigolosebastiano fromhomogeneousmetricspacestoliegroups AT ottazzialessandro fromhomogeneousmetricspacestoliegroups |