Commability of groups quasi-isometric to trees

Carette, Mathieu

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 $G\to H$ 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 $G\to H$ 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.

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

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