2022, 18: 161-208. doi: 10.3934/jmd.2022008

Eigenvalue gaps for hyperbolic groups and semigroups

1. 

CNRS and Laboratoire Alexander Grothendieck, Institut des Hautes Études Scientifiques, Université Paris-Saclay, 35 route de Chartres, 91440 Bures-sur-Yvette, France

2. 

CMAT, Facultad de Ciencias, Universidad de la Repú-blica, Iguá 4225, Montevideo, 11400, Uruguay

Received  February 18, 2020 Revised  August 04, 2021

Fund Project: FK: Partially supported by the Louis D. Foundation. This project received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC starting grant DiGGeS, grant agreement No. 715982).
RP: Partially supported by CSIC-618, FCE-135352, FCE-148740 and MathAmSud. This work was completed while RP was a von Neumann fellow at IAS, funded by Minerva Research Foundation Membership Fund and NSF DMS-1638352.

Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the $ i^\text{th} $ and $ (i+1)^\text{th} $ Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index $ i $. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.

Citation: Fanny Kassel, Rafael Potrie. Eigenvalue gaps for hyperbolic groups and semigroups. Journal of Modern Dynamics, 2022, 18: 161-208. doi: 10.3934/jmd.2022008
References:
[1]

H. AbelsG. A. Margulis and G. A. Soifer, Semigroups containing proximal linear maps, Israel J. Math., 91 (1995), 1-30.  doi: 10.1007/BF02761637.

[2]

A. Avila and J. Bochi, Lyapunov Exponents, Trieste Lecture Notes, 2008, available at http://www.mat.uc.cl/ jairo.bochi/docs/trieste.pdf.

[3]

A. AvilaJ. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $ {\text{SL}} (2, \Bbb R)$-cocycles, Comment. Math. Helv., 85 (2010), 813-884.  doi: 10.4171/CMH/212.

[4]

Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47.  doi: 10.1007/PL00001613.

[5]

Y. Benoist, Sous-Groupes Discrets des Groupes de Lie, Notes from the European Summer School in Group Theory, Luminy, 1997, available at https://www.math.u-psud.fr/ benoist/prepubli/0097luminy.pdf.

[6]

Y. Benoist and J. F. Quint, Random Walks on Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 62, Springer, Cham, 2016.

[7]

J. Bochi, Ergodic optimization of Birkhoff averages and Lyapunov exponents, in Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific, 3 (2018), 1821–1842.

[8]

J. Bochi, Ergodic optimization of Birkhoff averages and Lyapunov exponents, slides of a talk at the ICM 2018, available at http://www.mat.uc.cl/ jairo.bochi/docs/optimization_LE_talk_ICM.pdf.

[9]

J. Bochi and E. Garibaldi, Extremal norms for fiber bunched cocycles, J. Éc. Polytech. Math., 6 (2019), 947-1004.  doi: 10.5802/jep.109.

[10]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231.  doi: 10.1007/s00209-009-0494-y.

[11]

J. Bochi and I. Morris, Equilibrium states of generalised singular value potentials and applications to affine iterated function systems, Geom. Funct. Anal., 28 (2018), 995-1028.  doi: 10.1007/s00039-018-0447-x.

[12]

J. BochiR. Potrie and A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc., 21 (2019), 3343-3414.  doi: 10.4171/JEMS/905.

[13]

J. Bochi and M. Rams, The entropy of Lyapunov-optimizing measures of some matrix cocycles, J. Mod. Dyn., 10 (2016), 255-286.  doi: 10.3934/jmd.2016.10.255.

[14]

A. Borel and J. Tits, Groupes réductifs, (French) Publ. Math. Inst. Hautes Études Sci., 27 (1965), 55-150. 

[15]

E. Breuillard and C. Sert, The joint spectrum, J. Lond. Math. Soc., 103 (2021), 943-990.  doi: 10.1112/jlms.12397.

[16]

M. BridgemanR. D. CanaryF. Labourie and A. Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal., 25 (2015), 1089-1179.  doi: 10.1007/s00039-015-0333-8.

[17]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.

[18]

J. Burillo and M. Elder, Metric properties of Baumslag–Solitar groups, Internat. J. Algebra Comput., 25 (2015), 799-811.  doi: 10.1142/S0218196715500198.

[19]

C. Butler, Characterizing symmetric spaces by their Lyapunov spectra, preprint, arXiv: 1709.08066v4.

[20]

M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et Théorie des Groupes. Les Groupes Hyperboliques de Gromov, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990.

[21]

S. Crovisier and R. Potrie, Introduction to Partial Hyperbolicity, Trieste Lecture Notes, 2015, available at http://www.cmat.edu.uy/ rpotrie/documentos/pdfs/Crovisier-Potrie-PH.pdf.

[22]

J. Danciger, F. Guéritaud and F. Kassel, Convex cocompact actions in real projective geometry, preprint, arXiv: 1704.08711.

[23]

M. Delzant, O. Guichard, F. Labourie and S. Mozes, Displacing representations and orbit maps, In Geometry, Rigidity, and Group Actions, University of Chicago Press, (2011), 494–514.

[24]

A. J. Duncan and R. H. Gilman, Word hyperbolic semigroups, Math. Proc. Camb. Phil. Soc., 136 (2004), 513-524.  doi: 10.1017/S0305004103007497.

[25]

D. B. A. Epstein and K. Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology, 36 (1997), 1275-1289.  doi: 10.1016/S0040-9383(96)00046-8.

[26]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.

[27]

J. Fountain and M. Kambites, Hyperbolic groups and completely simple semigroups, In Semigroups and Languages, (2004), 106–132. doi: 10.1142/9789812702616_0007.

[28]

R. H. Gilman, On the definition of word hyperbolic groups, Math. Z., 242 (2002), 529-541.  doi: 10.1007/s002090100356.

[29]

A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov, Ergodic Theory Dynam. Systems, 30 (2010), 441-456.  doi: 10.1017/S0143385709000169.

[30]

F. GuéritaudO. GuichardF. Kassel and A. Wienhard, Anosov representations and proper actions, Geom. Topol., 21 (2017), 485-584.  doi: 10.2140/gt.2017.21.485.

[31]

O. Guichard and A. Wienhard, Anosov representations: Domains of discontinuity and applications, Invent. Math., 190 (2012), 357-438.  doi: 10.1007/s00222-012-0382-7.

[32]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502.  doi: 10.1007/BF02564647.

[33]

J. M. Howie, Fundamentals of Semi-Group Theory, London Mathematical Society Monographs, New Series, vol. 12, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[34]

B. Kalinin, Livšic theorem for matrix cocycles, Ann. of Math., 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.

[35]

I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and Geometric Group Theory, Contemp. Math., Amer. Math. Soc., Providence, RI, 296 (2002), 39–94. doi: 10.1090/conm/296/05068.

[36]

M. KapovichB. Leeb and J. Porti, Some recent results on Anosov representations, Transform. Groups, 21 (2016), 1105-1121.  doi: 10.1007/s00031-016-9393-6.

[37]

M. KapovichB. Leeb and J. Porti, A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol., 22 (2018), 3827-3923.  doi: 10.2140/gt.2018.22.3827.

[38]

F. Kassel, Proper actions on corank-one reductive homogeneous spaces, J. Lie Theory, 18 (2008), 961-978. 

[39]

F. Kassel, Geometric structures and representations of discrete groups, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, World Scientific, 2 (2018), 1113–1150.

[40]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[41]

A. Knapp, Lie Groups Beyond an Introduction, 2$^{nd}$ edition, Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002.

[42]

F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114.  doi: 10.1007/s00222-005-0487-3.

[43] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[44]

G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math., 78 (1956), 200-221.  doi: 10.2307/2372490.

[45]

D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math., 172 (2010), 1-39.  doi: 10.4007/annals.2010.172.1.

[46]

K. Park, Quasi-multiplicativity of typical cocycles, Comm. Math. Phys., 376 (2020), 1957-2004.  doi: 10.1007/s00220-020-03701-8.

[47]

R. Potrie, Robust dynamics, invariant structures and topological classification, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, World Scientific, 3 (2018), 2057–2080.

[48]

R. Velozo Ruiz, Characterization of uniform hyperbolicity for fibre-bunched cocycles, Dyn. Syst., 35 (2020), 124-139.  doi: 10.1080/14689367.2019.1649364.

[49]

M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, vol. 145, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139976602.

[50]

B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474.  doi: 10.1007/BF01295322.

[51]

A. Wienhard, An invitation to higher Teichmüller theory, in Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018, World Scientific, 2 (2018), 1007–1034.

[52]

J.-C. Yoccoz, Some questions and remarks on $ \mathrm{SL}(2, {{\mathbb R}})$ cocycles, in Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, (2004), 447–458.

[53]

A. Zimmer, Projective Anosov representations, convex cocompact actions, and rigidity, J. Differential Geom., 119 (2021), 513-586.  doi: 10.4310/jdg/1635368438.

show all references

References:
[1]

H. AbelsG. A. Margulis and G. A. Soifer, Semigroups containing proximal linear maps, Israel J. Math., 91 (1995), 1-30.  doi: 10.1007/BF02761637.

[2]

A. Avila and J. Bochi, Lyapunov Exponents, Trieste Lecture Notes, 2008, available at http://www.mat.uc.cl/ jairo.bochi/docs/trieste.pdf.

[3]

A. AvilaJ. Bochi and J.-C. Yoccoz, Uniformly hyperbolic finite-valued $ {\text{SL}} (2, \Bbb R)$-cocycles, Comment. Math. Helv., 85 (2010), 813-884.  doi: 10.4171/CMH/212.

[4]

Y. Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal., 7 (1997), 1-47.  doi: 10.1007/PL00001613.

[5]

Y. Benoist, Sous-Groupes Discrets des Groupes de Lie, Notes from the European Summer School in Group Theory, Luminy, 1997, available at https://www.math.u-psud.fr/ benoist/prepubli/0097luminy.pdf.

[6]

Y. Benoist and J. F. Quint, Random Walks on Reductive Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, vol. 62, Springer, Cham, 2016.

[7]

J. Bochi, Ergodic optimization of Birkhoff averages and Lyapunov exponents, in Proceedings of the International Congress of Mathematicians–Rio de Janeiro 2018, World Scientific, 3 (2018), 1821–1842.

[8]

J. Bochi, Ergodic optimization of Birkhoff averages and Lyapunov exponents, slides of a talk at the ICM 2018, available at http://www.mat.uc.cl/ jairo.bochi/docs/optimization_LE_talk_ICM.pdf.

[9]

J. Bochi and E. Garibaldi, Extremal norms for fiber bunched cocycles, J. Éc. Polytech. Math., 6 (2019), 947-1004.  doi: 10.5802/jep.109.

[10]

J. Bochi and N. Gourmelon, Some characterizations of domination, Math. Z., 263 (2009), 221-231.  doi: 10.1007/s00209-009-0494-y.

[11]

J. Bochi and I. Morris, Equilibrium states of generalised singular value potentials and applications to affine iterated function systems, Geom. Funct. Anal., 28 (2018), 995-1028.  doi: 10.1007/s00039-018-0447-x.

[12]

J. BochiR. Potrie and A. Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc., 21 (2019), 3343-3414.  doi: 10.4171/JEMS/905.

[13]

J. Bochi and M. Rams, The entropy of Lyapunov-optimizing measures of some matrix cocycles, J. Mod. Dyn., 10 (2016), 255-286.  doi: 10.3934/jmd.2016.10.255.

[14]

A. Borel and J. Tits, Groupes réductifs, (French) Publ. Math. Inst. Hautes Études Sci., 27 (1965), 55-150. 

[15]

E. Breuillard and C. Sert, The joint spectrum, J. Lond. Math. Soc., 103 (2021), 943-990.  doi: 10.1112/jlms.12397.

[16]

M. BridgemanR. D. CanaryF. Labourie and A. Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal., 25 (2015), 1089-1179.  doi: 10.1007/s00039-015-0333-8.

[17]

M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-12494-9.

[18]

J. Burillo and M. Elder, Metric properties of Baumslag–Solitar groups, Internat. J. Algebra Comput., 25 (2015), 799-811.  doi: 10.1142/S0218196715500198.

[19]

C. Butler, Characterizing symmetric spaces by their Lyapunov spectra, preprint, arXiv: 1709.08066v4.

[20]

M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et Théorie des Groupes. Les Groupes Hyperboliques de Gromov, Lecture Notes in Mathematics, vol. 1441, Springer-Verlag, Berlin, 1990.

[21]

S. Crovisier and R. Potrie, Introduction to Partial Hyperbolicity, Trieste Lecture Notes, 2015, available at http://www.cmat.edu.uy/ rpotrie/documentos/pdfs/Crovisier-Potrie-PH.pdf.

[22]

J. Danciger, F. Guéritaud and F. Kassel, Convex cocompact actions in real projective geometry, preprint, arXiv: 1704.08711.

[23]

M. Delzant, O. Guichard, F. Labourie and S. Mozes, Displacing representations and orbit maps, In Geometry, Rigidity, and Group Actions, University of Chicago Press, (2011), 494–514.

[24]

A. J. Duncan and R. H. Gilman, Word hyperbolic semigroups, Math. Proc. Camb. Phil. Soc., 136 (2004), 513-524.  doi: 10.1017/S0305004103007497.

[25]

D. B. A. Epstein and K. Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology, 36 (1997), 1275-1289.  doi: 10.1016/S0040-9383(96)00046-8.

[26]

D. J. Feng, Equilibrium states for factor maps between subshifts, Adv. Math., 226 (2011), 2470-2502.  doi: 10.1016/j.aim.2010.09.012.

[27]

J. Fountain and M. Kambites, Hyperbolic groups and completely simple semigroups, In Semigroups and Languages, (2004), 106–132. doi: 10.1142/9789812702616_0007.

[28]

R. H. Gilman, On the definition of word hyperbolic groups, Math. Z., 242 (2002), 529-541.  doi: 10.1007/s002090100356.

[29]

A. Gogolev, Diffeomorphisms Hölder conjugate to Anosov, Ergodic Theory Dynam. Systems, 30 (2010), 441-456.  doi: 10.1017/S0143385709000169.

[30]

F. GuéritaudO. GuichardF. Kassel and A. Wienhard, Anosov representations and proper actions, Geom. Topol., 21 (2017), 485-584.  doi: 10.2140/gt.2017.21.485.

[31]

O. Guichard and A. Wienhard, Anosov representations: Domains of discontinuity and applications, Invent. Math., 190 (2012), 357-438.  doi: 10.1007/s00222-012-0382-7.

[32]

M. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2, Comment. Math. Helv., 58 (1983), 453-502.  doi: 10.1007/BF02564647.

[33]

J. M. Howie, Fundamentals of Semi-Group Theory, London Mathematical Society Monographs, New Series, vol. 12, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.

[34]

B. Kalinin, Livšic theorem for matrix cocycles, Ann. of Math., 173 (2011), 1025-1042.  doi: 10.4007/annals.2011.173.2.11.

[35]

I. Kapovich and N. Benakli, Boundaries of hyperbolic groups, in Combinatorial and Geometric Group Theory, Contemp. Math., Amer. Math. Soc., Providence, RI, 296 (2002), 39–94. doi: 10.1090/conm/296/05068.

[36]

M. KapovichB. Leeb and J. Porti, Some recent results on Anosov representations, Transform. Groups, 21 (2016), 1105-1121.  doi: 10.1007/s00031-016-9393-6.

[37]

M. KapovichB. Leeb and J. Porti, A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol., 22 (2018), 3827-3923.  doi: 10.2140/gt.2018.22.3827.

[38]

F. Kassel, Proper actions on corank-one reductive homogeneous spaces, J. Lie Theory, 18 (2008), 961-978. 

[39]

F. Kassel, Geometric structures and representations of discrete groups, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, World Scientific, 2 (2018), 1113–1150.

[40]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.

[41]

A. Knapp, Lie Groups Beyond an Introduction, 2$^{nd}$ edition, Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002.

[42]

F. Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math., 165 (2006), 51-114.  doi: 10.1007/s00222-005-0487-3.

[43] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[44]

G. D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math., 78 (1956), 200-221.  doi: 10.2307/2372490.

[45]

D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math., 172 (2010), 1-39.  doi: 10.4007/annals.2010.172.1.

[46]

K. Park, Quasi-multiplicativity of typical cocycles, Comm. Math. Phys., 376 (2020), 1957-2004.  doi: 10.1007/s00220-020-03701-8.

[47]

R. Potrie, Robust dynamics, invariant structures and topological classification, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018, World Scientific, 3 (2018), 2057–2080.

[48]

R. Velozo Ruiz, Characterization of uniform hyperbolicity for fibre-bunched cocycles, Dyn. Syst., 35 (2020), 124-139.  doi: 10.1080/14689367.2019.1649364.

[49]

M. Viana, Lectures on Lyapunov Exponents, Cambridge Studies in Advanced Mathematics, vol. 145, Cambridge University Press, Cambridge, 2014. doi: 10.1017/CBO9781139976602.

[50]

B. Weiss, Subshifts of finite type and sofic systems, Monatsh. Math., 77 (1973), 462-474.  doi: 10.1007/BF01295322.

[51]

A. Wienhard, An invitation to higher Teichmüller theory, in Proceedings of the International Congress of Mathematicians-Rio de Janeiro 2018, World Scientific, 2 (2018), 1007–1034.

[52]

J.-C. Yoccoz, Some questions and remarks on $ \mathrm{SL}(2, {{\mathbb R}})$ cocycles, in Modern Dynamical Systems and Applications, Cambridge University Press, Cambridge, (2004), 447–458.

[53]

A. Zimmer, Projective Anosov representations, convex cocompact actions, and rigidity, J. Differential Geom., 119 (2021), 513-586.  doi: 10.4310/jdg/1635368438.

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