no. 4
Commability of groups quasi-isometric to trees
[Commabilité des groupes quasi-isométriques à des arbres]
Carette, Mathieu1
1 Université catholique de Louvain IRMP Chemin du Cyclotron 2, bte L7.01.01 1348 Louvain-la-Neuve (Belgium)
Annales de l'Institut Fourier, Tome 68 (2018) no. 4, pp. 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 G→H 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 G→H 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 :
Accepté le :
Publié le :
DOI : 10.5802/aif.3190
Classification : 22D05, 20F65, 20E08, 20E42
Keywords: Commability, groups acting on trees, quasi-isometric rigidity
Mot clés : Commabilité, groupes agissant sur des arbres, rigidité quasi-isométrique

Carette, Mathieu 1

1 Université catholique de Louvain IRMP Chemin du Cyclotron 2, bte L7.01.01 1348 Louvain-la-Neuve (Belgium)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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/articles/10.5802/aif.3190/

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