no. 4
Commability of groups quasi-isometric to trees
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, Volume 68 (2018) no. 4, pp. 1365-1398.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/aif.3190
Classification: 22D05, 20F65, 20E08, 20E42
Keywords: Commability, groups acting on trees, quasi-isometric rigidity
Mots-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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Carette, Mathieu. Commability of groups quasi-isometric to trees. Annales de l'Institut Fourier, Volume 68 (2018) no. 4, pp. 1365-1398. doi : 10.5802/aif.3190. https://aif.centre-mersenne.org/articles/10.5802/aif.3190/

[1] Bass, Hyman Covering theory for graphs of groups, J. Pure Appl. Algebra, Volume 89 (1993) no. 1-2, pp. 3-47 | DOI | MR

[2] Bass, Hyman; Kulkarni, Ravi Uniform tree lattices, J. Am. Math. Soc., Volume 3 (1990) no. 4, pp. 843-902 | DOI | MR

[3] Bass, Hyman; Lubotzky, Alexander Rigidity of group actions on locally finite trees, Proc. Lond. Math. Soc., Volume 69 (1994) no. 3, pp. 541-575 | DOI | MR

[4] Caprace, Pierre-Emmanuel; Cornulier, Yves; Monod, Nicolas; Tessera, Romain Amenable hyperbolic groups, J. Eur. Math. Soc., Volume 17 (2015) no. 11, pp. 2903-2947 | Zbl

[5] Carette, Mathieu; Dreesen, Dennis Locally compact convergence groups and n-transitive actions, Math. Z., Volume 278 (2014) no. 3-4, pp. 795-827 | DOI | MR

[6] Carette, Mathieu; Tessera, Romain Geometric rigidity and flexibility for groups acting on trees (In preparation)

[7] Cornulier, Yves Commability and focal locally compact groups, Indiana Univ. Math. J., Volume 64 (2015) no. 1, pp. 115-150 | Zbl

[8] Cornulier, Yves On the quasi-isometric classification of focal hyperbolic groups, New Directions in locally compact groups (London Mathematical Society Lecture Note Series), Volume 447, Cambridge University Press, 2018, pp. 275-342 | DOI

[9] Forester, Max Deformation and rigidity of simplicial group actions on trees, Geom. Topol., Volume 6 (2002), p. 219-267 (electronic) | DOI | MR

[10] Kleiner, Bruce; Leeb, Bernhard Induced quasi-actions: a remark, Proc. Am. Math. Soc., Volume 137 (2009) no. 5, pp. 1561-1567 | DOI | MR

[11] Mosher, Lee; Sageev, Michah; Whyte, Kevin Maximally symmetric trees, Geom. Dedicata, Volume 92 (2002), pp. 195-233 (Dedicated to John Stallings on the occasion of his 65th birthday) | DOI | MR

[12] Mosher, Lee; Sageev, Michah; Whyte, Kevin Quasi-actions on trees. I. Bounded valence, Ann. Math., Volume 158 (2003) no. 1, pp. 115-164 | DOI | MR

[13] Serre, Jean-Pierre Trees, Springer, Berlin, 1980, ix+142 pages (Translated from the French by John Stillwell) | MR

[14] Tits, Jacques Sur le groupe des automorphismes d’un arbre, Essays on topology and related topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 188-211 | MR

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