no. 1
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
[Sur le rayon spectral joint pour les groupes d’isométries des espaces à courbure négative ou nulle et la croissance uniforme]
Breuillard, Emmanuel1 ; Fujiwara, Koji2
1 University of Cambridge, DPMMS Wilberforce Road CB3 0WB Cambridge (U.K.)
2 Department of Mathematics, Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502 (Japan)
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 317-391.

Nous généralisons la notion de rayon spectral joint dans le cadre des actions de groupes par isométries sur les espaces à courbure négative ou nulle et nous donnons des versions géométriques des résultats de Berger–Wang et de Bochi valables dans tout espace δ-hyperbolique ainsi que dans les espaces symétriques de type non-compact. Cette méthode permet de produire des éléments hyperboliques dans de nombreuses situations géométriques classiques. Nous donnons par ailleurs des applications à la croissance uniforme ainsi qu’une nouvelle preuve et une généralisation d’un théorème de Besson–Courtois–Gallot.

We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger–Wang and Bochi valid for δ-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson–Courtois–Gallot.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3374
Classification : 20F65, 20M20, 15A30
Keywords: joint spectral radius, $\mathrm{CAT}(0)$ spaces, Gromov hyperbolic spaces, uniform exponential growth, affine isometric actions, ping-pong
Mot clés : rayon spectral joint, espaces $\mathrm{CAT}(0)$, espaces hyperboliques au sens de Gromov, croissance exponentielle uniforme, actions isométriques affines, ping-pong
Breuillard, Emmanuel 1 ; Fujiwara, Koji 2

1 University of Cambridge, DPMMS Wilberforce Road CB3 0WB Cambridge (U.K.)
2 Department of Mathematics, Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_1_317_0,
     author = {Breuillard, Emmanuel and Fujiwara, Koji},
     title = {On the joint spectral radius for isometries of non-positively curved spaces and uniform growth},
     journal = {Annales de l'Institut Fourier},
     pages = {317--391},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {1},
     year = {2021},
     doi = {10.5802/aif.3374},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3374/}
}
TY  - JOUR
AU  - Breuillard, Emmanuel
AU  - Fujiwara, Koji
TI  - On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 317
EP  - 391
VL  - 71
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3374/
DO  - 10.5802/aif.3374
LA  - en
ID  - AIF_2021__71_1_317_0
ER  - 
%0 Journal Article
%A Breuillard, Emmanuel
%A Fujiwara, Koji
%T On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
%J Annales de l'Institut Fourier
%D 2021
%P 317-391
%V 71
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3374/
%R 10.5802/aif.3374
%G en
%F AIF_2021__71_1_317_0
Breuillard, Emmanuel; Fujiwara, Koji. On the joint spectral radius for isometries of non-positively curved spaces and uniform growth. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 317-391. doi : 10.5802/aif.3374. https://aif.centre-mersenne.org/articles/10.5802/aif.3374/

[1] Abels, Herbert; Margulis, Gregory Coarsely geodesic metrics on reductive groups, Modern dynamical systems and applications, Cambridge University Press, 2004, pp. 163-183 | MR | Zbl

[2] Arzhantseva, Goulnara N.; Burillo, J.; Lustig, Martin; Reeves, Laurence; Short, Hamish; Ventura, Enric Uniform non-amenability, Adv. Math., Volume 197 (2005) no. 2, pp. 499-522 | DOI | MR | Zbl

[3] Arzhantseva, Goulnara N.; Lysenok, Igor G. A lower bound on the growth of word hyperbolic groups, J. Lond. Math. Soc., Volume 73 (2006) no. 1, pp. 109-125 | DOI | MR | Zbl

[4] Bartholdi, Laurent; de Cornulier, Yves Infinite groups with large balls of torsion elements and small entropy, Arch. Math., Volume 87 (2006) no. 2, pp. 104-112 | DOI | MR | Zbl

[5] Bass, Hyman Groups of integral representation type, Pac. J. Math., Volume 86 (1980) no. 1, pp. 15-51 | DOI | MR | Zbl

[6] Berger, Marc A.; Wang, Yang Bounded semigroups of matrices, Linear Algebra Appl., Volume 166 (1992), pp. 21-27 | DOI | MR

[7] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre Uniform growth of groups acting on Cartan-Hadamard spaces, J. Eur. Math. Soc., Volume 13 (2011) no. 5, pp. 1343-1371 | DOI | MR

[8] Besson, Gérard; Courtois, Gilles; Gallot, Sylvestre; Sambusetti, Andrea Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces (2017) (https://arxiv.org/abs/1712.08386)

[9] Bestvina, Mladen ℝ-trees in topology, geometry, and group theory, Handbook of geometric topology, North-Holland, 2002, pp. 55-91 | MR

[10] Bochi, Jairo Inequalities for numerical invariants of sets of matrices, Linear Algebra Appl., Volume 368 (2003), pp. 71-81 | DOI | MR | Zbl

[11] Bochnak, Jacek; Coste, Michel; Roy, Marie-Françoise Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 36, Springer, 1998, x+430 pages (Translated from the 1987 French original, Revised by the authors) | DOI | MR | Zbl

[12] Borel, A. On free subgroups of semisimple groups, Enseign. Math., Volume 29 (1983) no. 1-2, pp. 151-164 | MR

[13] Bowditch, Brian H. A course on geometric group theory, MSJ Memoirs, 16, Mathematical Society of Japan, 2006, x+104 pages | DOI | MR

[14] Breuillard, Emmanuel Effective estimates for the spectral radius of a bounded set of matrices (unpublished note available upon request)

[15] Breuillard, Emmanuel A strong Tits alternative (2008) (https://arxiv.org/abs/0804.1395)

[16] Breuillard, Emmanuel A height gap theorem for finite subsets of GL d (ℚ ¯) and nonamenable subgroups, Ann. Math., Volume 174 (2011) no. 2, pp. 1057-1110 | DOI | MR | Zbl

[17] Breuillard, Emmanuel; Gelander, Tsachik Uniform independence in linear groups, Invent. Math., Volume 173 (2008) no. 2, pp. 225-263 | DOI | MR

[18] Breuillard, Emmanuel; Green, Ben; Guralnick, Robert; Tao, Terence Strongly dense free subgroups of semisimple algebraic groups, Isr. J. Math., Volume 192 (2012) no. 1, pp. 347-379 | DOI | MR | Zbl

[19] Breuillard, Emmanuel; Green, Ben; Tao, Terence The structure of approximate groups, Publ. Math., Inst. Hautes Étud. Sci., Volume 116 (2012), pp. 115-221 | DOI | Numdam | MR | Zbl

[20] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, 319, Springer, 1999, xxii+643 pages | DOI | MR | Zbl

[21] Burago, Yuriĭ D.; Zalgaller, Viktor A. Geometric inequalities, Grundlehren der Mathematischen Wissenschaften, 285, Springer, 1988, xiv+331 pages (Translated from the Russian by A. B. Sosinskiĭ) | DOI | MR

[22] Burger, Marc; Pozzetti, Maria Beatrice Maximal representations, non-Archimedean Siegel spaces, and buildings, Geom. Topol., Volume 21 (2017) no. 6, pp. 3539-3599 | DOI | MR | Zbl

[23] Caprace, Pierre-Emmanuel; Lytchak, Alexander At infinity of finite-dimensional CAT(0) spaces, Math. Ann., Volume 346 (2010) no. 1, pp. 1-21 | DOI | MR | Zbl

[24] Champetier, Christophe; Guirardel, Vincent Monoïdes libres dans les groupes hyperboliques, Séminaire de Théorie Spectrale et Géométrie, Vol. 18, Année 1999–2000 (Sémin. Théor. Spectr. Géom.), Volume 18, Université de Grenoble I, 2000, pp. 157-170 | DOI | Numdam | MR | Zbl

[25] Chepoi, Victor; Dragan, Feodor F.; Estellon, Bertrand; Habib, Michel; Vaxès, Yann Diameters, centers, and approximating trees of δ-hyperbolic geodesic spaces and graphs, Computational geometry (SCG’08), ACM Press, 2008, pp. 59-68 | DOI | MR | Zbl

[26] Chiswell, Ian Introduction to Λ-trees, World Scientific, 2001, xii+315 pages | DOI | MR | Zbl

[27] Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov, Lecture Notes in Mathematics, 1441, Springer, 1990, x+165 pages | MR | Zbl

[28] de Cornulier, Yves; Tessera, Romain; Valette, Alain Isometric group actions on Hilbert spaces: growth of cocycles, Geom. Funct. Anal., Volume 17 (2007) no. 3, pp. 770-792 | DOI | MR | Zbl

[29] Daubechies, Ingrid; Lagarias, Jeffrey C. Sets of matrices all infinite products of which converge, Linear Algebra Appl., Volume 161 (1992), pp. 227-263 | DOI | MR | Zbl

[30] Delzant, Thomas Sous-groupes distingués et quotients des groupes hyperboliques, Duke Math. J., Volume 83 (1996) no. 3, pp. 661-682 | DOI | MR | Zbl

[31] Delzant, Thomas; Steenbock, Markus Product set growth in groups and hyperbolic geometry (2018) (https://arxiv.org/abs/1804.01867) | Zbl

[32] Eskin, Alex; Mozes, Shahar; Oh, Hee On uniform exponential growth for linear groups, Invent. Math., Volume 160 (2005) no. 1, pp. 1-30 | DOI | MR | Zbl

[33] Farb, Benson Group actions and Helly’s theorem, Adv. Math., Volume 222 (2009) no. 5, pp. 1574-1588 | DOI | MR | Zbl

[34] Fujiwara, Koji Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents, Trans. Am. Math. Soc., Volume 367 (2015) no. 6, pp. 4377-4405 | DOI | MR | Zbl

[35] Sur les groupes hyperboliques d’après Mikhael Gromov (Ghys, É.; de la Harpe, P., eds.), Progress in Mathematics, 83, Birkhäuser, 1990, xii+285 pages (Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988) | DOI | MR | Zbl

[36] Grigorchuk, Rostislav; de la Harpe, Pierre Limit behaviour of exponential growth rates for finitely generated groups, Essays on geometry and related topics, Vol. 1, 2 (Monographies de l’Enseignement Mathématique), Volume 38, L’Enseignement Mathématique, 2001, pp. 351-370 | MR | Zbl

[37] Gromov, M. Hyperbolic groups, Essays in group theory (Mathematical Sciences Research Institute Publications), Volume 8, Springer, 1987, pp. 75-263 | DOI | MR | Zbl

[38] Gromov, Mikhail Random walk in random groups, Geom. Funct. Anal., Volume 13 (2003) no. 1, pp. 73-146 | DOI | MR | Zbl

[39] Gromov, Mikhail; Schoen, Richard Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math., Inst. Hautes Étud. Sci. (1992) no. 76, pp. 165-246 | DOI | Numdam | MR | Zbl

[40] Guralnick, Robert Some applications of subgroup structure to probabilistic generation and covers of curves, Algebraic groups and their representations (Cambridge, 1997) (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 517, Kluwer Academic Publishers, 1998, pp. 301-320 | DOI | MR | Zbl

[41] de la Harpe, Pierre; Bucher, Michèle Free products with amalgamation, and HNN-extensions of uniformly exponential growth, Mat. Zametki, Volume 67 (2000) no. 6, pp. 811-815 | DOI | MR | Zbl

[42] Kabenyuk, Mikhaĭl I. Compact elements and Cartan subgroups of connected Lie groups, Ukr. Mat. Zh., Volume 42 (1990) no. 2, pp. 164-168 | DOI | MR | Zbl

[43] Kabenyuk, Mikhaĭl I. Compact elements in connected Lie groups, J. Lie Theory, Volume 27 (2017) no. 2, pp. 569-578 | MR | Zbl

[44] Kapovich, Michael; Leeb, Bernhard Discrete isometry groups of symmetric spaces, Handbook of group actions. Vol. IV (Advanced Lectures in Mathematics (ALM)), Volume 41, International Press, 2018, pp. 191-290 | MR | Zbl

[45] Kar, Aditi; Sageev, Michah Uniform exponential growth for CAT(0) square complexes, Algebr. Geom. Topol., Volume 19 (2019) no. 3, pp. 1229-1245 | DOI | MR | Zbl

[46] Karlsson, Anders; Margulis, Gregory A multiplicative ergodic theorem and nonpositively curved spaces, Commun. Math. Phys., Volume 208 (1999) no. 1, pp. 107-123 | DOI | MR | Zbl

[47] Kleiner, Bruce The local structure of length spaces with curvature bounded above, Math. Z., Volume 231 (1999) no. 3, pp. 409-456 | DOI | MR | Zbl

[48] Korevaar, Nicholas J.; Schoen, Richard Sobolev spaces and harmonic maps for metric space targets, Commun. Anal. Geom., Volume 1 (1993) no. 3-4, pp. 561-659 | DOI | MR | Zbl

[49] Kostant, Bertram On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. Éc. Norm. Supér., Volume 6 (1974), pp. 413-455 | DOI | MR | Zbl

[50] Koubi, Malik Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier, Volume 48 (1998) no. 5, pp. 1441-1453 | DOI | Numdam | MR | Zbl

[51] Lafforgue, Vincent Une remarque sur les fonctions conditionnellement de type négatif, C. R. Math. Acad. Sci. Paris, Volume 342 (2006) no. 7, pp. 497-499 | DOI | MR | Zbl

[52] Larsen, Michael Word maps have large image, Isr. J. Math., Volume 139 (2004), pp. 149-156 | DOI | MR | Zbl

[53] Le Boudec, Adrien Groups acting on trees with almost prescribed local action, Comment. Math. Helv., Volume 91 (2016) no. 2, pp. 253-293 | DOI | MR | Zbl

[54] Mann, Avinoam The growth of free products, J. Algebra, Volume 326 (2011), pp. 208-217 | DOI | MR | Zbl

[55] Monod, Nicolas Superrigidity for irreducible lattices and geometric splitting, J. Am. Math. Soc., Volume 19 (2006) no. 4, pp. 781-814 | DOI | MR | Zbl

[56] Oregón-Reyes, Eduardo Properties of sets of isometries of Gromov hyperbolic spaces, Groups Geom. Dyn., Volume 12 (2018) no. 3, pp. 889-910 | DOI | MR | Zbl

[57] Osin, Denis V. Algebraic entropy of elementary amenable groups, Geom. Dedicata, Volume 107 (2004), pp. 133-151 | DOI | MR | Zbl

[58] Papadima, Ştefan Discrete symmetry, toral symmetry and the Euler characteristic of manifolds, Proc. Am. Math. Soc., Volume 103 (1988) no. 2, pp. 612-614 | DOI | MR | Zbl

[59] Parreau, Anne La distance vectorielle dans les immeubles affines et les espaces symétriques (preprint)

[60] Parreau, Anne Sous-groupes elliptiques de groupes linéaires sur un corps valué, J. Lie Theory, Volume 13 (2003) no. 1, pp. 271-278 | MR | Zbl

[61] Raghunathan, Madabusi S. Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 68, Springer, 1972, ix+227 pages | MR

[62] Rota, Gian-Carlo; Strang, Gilbert A note on the joint spectral radius, Nederl. Akad. Wet., Proc., Ser. A, Volume 63 (1960), pp. 379-381 | MR | Zbl

[63] Schur, Issai Uber Gruppen periodischer Substitutionen, Sitzber. Preuss. Akad. Wiss., Volume 1911 (1911), pp. 619-627 | Zbl

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

[65] Shalom, Yehuda Rigidity of commensurators and irreducible lattices, Invent. Math., Volume 141 (2000) no. 1, pp. 1-54 | DOI | MR | Zbl

Cité par Sources :