Commability of groups quasi-isometric to trees  [ Commabilité des groupes quasi-isométriques à des arbres ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 4, p. 1365-1398
La commabilité est la relation d’équivalence entre groupes localement compacts la plus fine telle que G et H sont équivalents dès qu’il existe un homomorphisme GH continu, propre et d’image cocompacte. Répondant à une question de Cornulier, nous montrons que tous les groupes localement compacts non-élémentaires agissant sur des arbres simpliciaux localement finis sont commables, renforçant les formes précédentes de rigidité quasi-isométrique pour les arbres. De plus, nous montrons que 6 homomorphismes suffisent toujours, et donnons le premier exemple d’une paire de groupes localement compacts qui sont commables mais n’ayant pas de commation constituée de moins de 6 homomorphismes. Notre rigidité quasi-isométrique forte s’applique également à des produits d’espace symétriques et d’immeubles euclidiens, dont certains facteurs sont éventuellement des arbres.
Commability is the finest equivalence relation between locally compact groups such that G and H are equivalent whenever there is a continuous proper homomorphism GH with cocompact image. Answering a question of Cornulier, we show that all non-elementary locally compact groups acting geometrically on locally finite simplicial trees are commable, thereby strengthening previous forms of quasi-isometric rigidity for trees. We further show that 6 homomorphisms always suffice, and provide the first example of a pair of locally compact groups which are commable but without commation consisting of less than 6 homomorphisms. Our strong quasi-isometric rigidity also applies to products of symmetric spaces and Euclidean buildings, possibly with some factors being trees.
Reçu le : 2013-11-28
Accepté le : 2014-09-30
Publié le : 2018-11-23
DOI : https://doi.org/10.5802/aif.3190
Classification:  22D05,  20F65,  20E08,  20E42
Mots clés: Commabilité, groupes agissant sur des arbres, rigidité quasi-isométrique
@article{AIF_2018__68_4_1365_0,
     author = {Carette, Mathieu},
     title = {Commability of groups quasi-isometric to trees},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {4},
     year = {2018},
     pages = {1365-1398},
     doi = {10.5802/aif.3190},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_4_1365_0}
}
Carette, Mathieu. Commability of groups quasi-isometric to trees. Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 1365-1398. doi : 10.5802/aif.3190. https://aif.centre-mersenne.org/item/AIF_2018__68_4_1365_0/

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