Relatively dominated representations
Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2169-2235.

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness. We prove that groups admitting relatively dominated representations must be relatively hyperbolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich–Leeb which also introduces higher-rank analogues of geometric finiteness.

Les représentations Anosov fournissent une classe de sous-groupes discrets des groupes de Lie qui généralisent les sous-groupes convexes-cocompacts d’un groupe de Lie de rang un. En rang un, la classe des sous-groupes géométriquement finis est une généralisation de la classe des sous-groupes convexes-cocompacts, qui autorise des défauts isolés d’hyperbolicité. Nous introduisons les représentations relativement dominées comme une relativisation des représentations Anosov, autrement dit un analogue de la finitude géométrique en rang supérieur. Nous montrons qu’un groupe qui admet une représentation relativement dominée est nécessairement relativement hyperbolique et que ces représentations induisent des applications de bord satisfaisant des bonnes propriétés. Nous donnons des exemples et faisons des connexions avec le travail de Kapovich–Leeb sur d’ autres analogues de la finitude géometrique en rang supérieur.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/aif.3449
Classification: 22E40,  20F67,  37D30,  53C35
Keywords: Discrete subgroups of Lie groups, geometric finiteness, dominated splittings, relatively hyperbolic groups
Zhu, Feng 1

1 Faculty of Mathematics, Technion, 733 Amado Building, 3200003 Haifa, (Israel)
License: CC-BY-ND 4.0
@article{AIF_2021__71_5_2169_0,
     author = {Zhu, Feng},
     title = {Relatively dominated representations},
     journal = {Annales de l'Institut Fourier},
     pages = {2169--2235},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {5},
     year = {2021},
     doi = {10.5802/aif.3449},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3449/}
}
TY  - JOUR
TI  - Relatively dominated representations
JO  - Annales de l'Institut Fourier
PY  - 2021
DA  - 2021///
SP  - 2169
EP  - 2235
VL  - 71
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3449/
UR  - https://doi.org/10.5802/aif.3449
DO  - 10.5802/aif.3449
LA  - en
ID  - AIF_2021__71_5_2169_0
ER  - 
%0 Journal Article
%T Relatively dominated representations
%J Annales de l'Institut Fourier
%D 2021
%P 2169-2235
%V 71
%N 5
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.3449
%R 10.5802/aif.3449
%G en
%F AIF_2021__71_5_2169_0
Zhu, Feng. Relatively dominated representations. Annales de l'Institut Fourier, Volume 71 (2021) no. 5, pp. 2169-2235. doi : 10.5802/aif.3449. https://aif.centre-mersenne.org/articles/10.5802/aif.3449/

[1] Abels, Herbert; Margulis, Gregory A.; Soĭfer, Grigoriĭ A. Semigroups containing proximal linear maps, Isr. J. Math., Volume 91 (1995) no. 1-3, pp. 1-30 | DOI | MR | Zbl

[2] Benoist, Yves Propriétés Asymptotiques des Groupes Linéaires, Geom. Funct. Anal., Volume 7 (1997) no. 1, pp. 1-47 | DOI | Zbl

[3] Bochi, Jairo; Gourmelon, Nicolas Some characterizations of domination, Math. Z., Volume 263 (2009) no. 1, pp. 221-231 | DOI | MR | Zbl

[4] Bochi, Jairo; Potrie, Rafael; Sambarino, Andrés Anosov representations and dominated splittings, J. Eur. Math. Soc., Volume 21 (2019) no. 11, pp. 3343-3414 | DOI | MR | Zbl

[5] Bowditch, Brian H. A Topological Characterisation of Hyperbolic Groups, J. Am. Math. Soc., Volume 11 (1998) no. 3, pp. 643-667 | DOI | MR | Zbl

[6] Bowditch, Brian H. Relatively hyperbolic groups, Int. J. Algebra Comput., Volume 22 (2012) no. 03, 1250016 | DOI | MR | Zbl

[7] Bridgeman, Martin; Canary, Richard; Labourie, François; Sambarino, Andrés The pressure metric for Anosov representations, Geom. Funct. Anal., Volume 25 (2015) no. 4, pp. 1089-1179 | DOI | MR | Zbl

[8] Cannon, James W.; Cooper, Daryl A Characterization of Cocompact Hyperbolic and Finite-Volume Hyperbolic Groups in Dimension Three, Trans. Am. Math. Soc., Volume 330 (1992) no. 1, pp. 419-431 | DOI | MR | Zbl

[9] Crampon, Mickaël; Marquis, Ludovic Finitude géométrique en géométrie de Hilbert, Ann. Inst. Fourier, Volume 64 (2014) no. 6, pp. 2299-2377 | DOI | Numdam | Zbl

[10] Crampon, Mickaël; Marquis, Ludovic Le flot géodésique des quotients géométriquement finis des géométries de Hilbert, Pac. J. Math., Volume 268 (2014) no. 2, pp. 313-369 | DOI | Zbl

[11] Farb, Benson Relatively hyperbolic groups, Geom. Funct. Anal., Volume 8 (1998) no. 5, pp. 810-840 | DOI | MR | Zbl

[12] Gerasimov, Victor Expansive Convergence Groups are Relatively Hyperbolic, Geom. Funct. Anal., Volume 19 (2009) no. 1, pp. 137-169 | DOI | MR | Zbl

[13] Groff, Bradley W. Quasi-isometries, boundaries and JSJ-decompositions of relatively hyperbolic groups, J. Topol. Anal., Volume 5 (2013) no. 4, pp. 451-475 | DOI | MR | Zbl

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

[15] Groves, Daniel; Manning, Jason F. Dehn filling in relatively hyperbolic groups, Isr. J. Math., Volume 168 (2008) no. 1, p. 317 | DOI | MR | Zbl

[16] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna Anosov representations and proper actions, Geom. Topol., Volume 21 (2017) no. 1, pp. 485-584 | DOI | MR | Zbl

[17] Guichard, Olivier; Wienhard, Anna Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | DOI | MR | Zbl

[18] Kapovich, Michael; Leeb, Bernhard Relativizing characterizations of Anosov subgroups. I (2018) (https://arxiv.org/abs/1807.00160)

[19] Kapovich, Michael; Leeb, Bernhard; Porti, Joan Some recent results on Anosov representations, Transform. Groups, Volume 21 (2016) no. 4, pp. 1105-1121 | DOI | MR | Zbl

[20] Kapovich, Michael; Leeb, Bernhard; Porti, Joan A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol., Volume 22 (2018) no. 7, pp. 3827-3923 | DOI | MR | Zbl

[21] Labourie, François Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114 | DOI | MR | Zbl

[22] Osin, Denis V. Relatively Hyperbolic Groups: Intrinsic Geometry, Algebraic Properties, and Algorithmic Problems, Memoirs of the American Mathematical Society, 843, American Mathematical Society, 2006 | MR | Zbl

[23] Quas, Anthony; Thieullen, Philippe; Zarrabi, Mohamed Explicit bounds for separation between Oseledets subspaces, Dyn. Syst., Volume 34 (2019) no. 3, pp. 517-560 | DOI | MR | Zbl

[24] Raghunathan, Madabusi S. A proof of Oseledec’s multiplicative ergodic theorem, Isr. J. Math., Volume 32 (1979) no. 4, pp. 356-362 | DOI | MR | Zbl

[25] Yaman, Asli A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math., Volume 566 (2004), pp. 41-89 | DOI | MR | Zbl

Cited by Sources: