[Les formes orthogonales des groupes de Kac–Moody sont à hyperbolicité acylindrique]
Nous fournissons des conditions suffisantes garantissant qu’un groupe donné opérant par isométries sur un espace métrique géodésique soit à hyperbolicité acylindrique. Diverses applications aux groupes d’isométries d’espaces CAT(0) sont mentionnées. Nous montrons en outre qu’un groupe d’automorphismes d’un immeuble irréductible non-sphérique et non-affine est à hyperbolicité acylindrique s’il existe une chambre à stabilisateur fini dont l’orbite contienne un appartement. Ce critère est finalement appliqué aux formes orthogonales des groupes de Kac–Moody sur des corps arbitraires. Il s’applique également aux produits graphés irréductibles de groupes arbitraires, ce qui fournit une nouvelle démonstration d’un résultat récent de Minasyan–Osin.
We give sufficient conditions for a group acting on a geodesic metric space to be acylindrically hyperbolic and mention various applications to groups acting on CAT() spaces. We prove that a group acting on an irreducible non-spherical non-affine building is acylindrically hyperbolic provided there is a chamber with finite stabiliser whose orbit contains an apartment. Finally, we show that the following classes of groups admit an action on a building with those properties: orthogonal forms of Kac–Moody groups over arbitrary fields, and irreducible graph products of arbitrary groups - recovering a result of Minasyan–Osin.
Keywords: hyperbolicity, acylindrical hyperbolicity, buildings, Kac–Moody groups
Mot clés : hyperbolicité, hyperbolicité acylindrique, immeubles, groupes de Kac–Moody
Caprace, Pierre-Emmanuel 1 ; Hume, David 1
@article{AIF_2015__65_6_2613_0,
author = {Caprace, Pierre-Emmanuel and Hume, David},
title = {Orthogonal forms of {Kac{\textendash}Moody} groups are acylindrically hyperbolic},
journal = {Annales de l'Institut Fourier},
pages = {2613--2640},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {65},
number = {6},
year = {2015},
doi = {10.5802/aif.2998},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2998/}
}
TY - JOUR AU - Caprace, Pierre-Emmanuel AU - Hume, David TI - Orthogonal forms of Kac–Moody groups are acylindrically hyperbolic JO - Annales de l'Institut Fourier PY - 2015 SP - 2613 EP - 2640 VL - 65 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2998/ DO - 10.5802/aif.2998 LA - en ID - AIF_2015__65_6_2613_0 ER -
%0 Journal Article %A Caprace, Pierre-Emmanuel %A Hume, David %T Orthogonal forms of Kac–Moody groups are acylindrically hyperbolic %J Annales de l'Institut Fourier %D 2015 %P 2613-2640 %V 65 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2998/ %R 10.5802/aif.2998 %G en %F AIF_2015__65_6_2613_0
Caprace, Pierre-Emmanuel; Hume, David. Orthogonal forms of Kac–Moody groups are acylindrically hyperbolic. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2613-2640. doi : 10.5802/aif.2998. https://aif.centre-mersenne.org/articles/10.5802/aif.2998/
[1] Buildings, Graduate Texts in Mathematics, 248, Springer, New York, 2008, pp. xxii+747 (Theory and applications) | DOI | MR | Zbl
[2] Lectures on spaces of nonpositive curvature, DMV Seminar, 25, Birkhäuser Verlag, Basel, 1995, pp. viii+112 (With an appendix by Misha Brin) | DOI | MR
[3] Bounded cohomology with coefficients in uniformly convex Banach spaces (http://arxiv.org/abs/1306.1542)
[4] Constructing group actions on quasi-trees and applications to mapping class groups (http://arxiv.org/abs/1006.1939, to appear in Publ. Math. IHES) | MR
[5] A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal., Volume 19 (2009) no. 1, pp. 11-40 | DOI | MR | Zbl
[6] Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, pp. xxii+643 | DOI | MR | Zbl
[7] Extended invariance of 11-dimensional supergravity, J. High Energy Phys. (2006) no. 2, pp. 056, 11 pp. (electronic) | DOI
[8] Normal subgroups in the Cremona group, Acta Math., Volume 210 (2013) no. 1, pp. 31-94 (With an appendix by Yves de Cornulier) | DOI | MR | Zbl
[9] Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal., Volume 19 (2010) no. 5, pp. 1296-1319 | DOI | MR | Zbl
[10] Open subgroups of locally compact Kac-Moody groups, Math. Z., Volume 274 (2013) no. 1-2, pp. 291-313 | DOI | MR | Zbl
[11] Isometry groups of non-positively curved spaces: discrete subgroups, J. Topol., Volume 2 (2009) no. 4, pp. 701-746 | DOI | MR | Zbl
[12] Simplicity and superrigidity of twin building lattices, Invent. Math., Volume 176 (2009) no. 1, pp. 169-221 | DOI | MR | Zbl
[13] Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal., Volume 21 (2011) no. 4, pp. 851-891 | DOI | MR | Zbl
[14] Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces (http://arxiv.org/abs/1111.7048, to appear in Mem. Amer. Math. Soc.)
[15] Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B, Volume 634 (2006) no. 2-3, pp. 319-324 | DOI | MR | Zbl
[16] Buildings are , Geometry and cohomology in group theory (Durham, 1994) (London Math. Soc. Lecture Note Ser.), Volume 252, Cambridge Univ. Press, Cambridge, 1998, pp. 108-123 | DOI | MR | Zbl
[17] Spin covers of maximal compact subgroups of Kac-Moody groups and spin extended Weyl groups (http://arxiv.org/abs/1502.07294)
[18] Lattices from involutions of Kac-Moody groups, Oberwolfach Rep., Volume 5 (2007), pp. 139-140
[19] Generalized spin representations (http://arxiv.org/abs/1110.5576, to appear in Münster J. of Math.)
[20] On topological twin buildings and topological split Kac-Moody groups, Innov. Incidence Geom., Volume 13 (2013), pp. 1-71 | MR | Zbl
[21] Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001, pp. xxvi+641 (Corrected reprint of the 1978 original) | DOI | MR | Zbl
[22] Embedding mapping class groups into finite products of trees (http://arxiv.org/abs/1207.2132)
[23] Abstract simplicity of locally compact Kac-Moody groups, Compos. Math., Volume 150 (2014) no. 4, pp. 713-728 | DOI | MR
[24] Conjugacy classes and straight elements in Coxeter groups, J. Algebra, Volume 407 (2014), pp. 68-80 | DOI | MR | Zbl
[25] Acylindrical hyperbolicity of groups acting on trees (http://arxiv.org/abs/1310.6289, to appear in Math. Annalen) | MR
[26] Coxeter groups act on cube complexes, J. Group Theory, Volume 6 (2003) no. 3, pp. 399-413 | DOI | MR | Zbl
[27] Strong Tits alternative for subgroups of Coxeter groups, J. Lie Theory, Volume 12 (2002) no. 1, pp. 259-264 | MR | Zbl
[28] Acylindrically hyperbolic groups (http://arxiv.org/abs/1304.1246, to appear in Trans. Amer. Math. Soc.) | MR
[29] Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. Paris Sér. I Math., Volume 329 (1999) no. 6, pp. 475-478 | DOI | MR | Zbl
[30] Contracting elements and random walks (http://arxiv.org/abs/1112.2666)
[31] Quasi-convexity of hyperbolically embedded subgroups (http://arxiv.org/abs/1310.7753, to appear in Math. Z.)
[32] Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra, Volume 105 (1987) no. 2, pp. 542-573 | DOI | MR | Zbl
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