Anti-de Sitter strictly GHC-regular groups which are not lattices

Lee, Gye-Seon; Marquis, Ludovic

doi:10.1090/tran/7530

Anti-de Sitter strictly GHC-regular groups which are not lattices
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by Gye-Seon Lee and Ludovic Marquis PDF

Trans. Amer. Math. Soc. 372 (2019), 153-186 Request permission

Abstract:

For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm {AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb {H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2 in a work of Barbot et al. and disproves Conjecture 8.11 of Barbot–Mérigot [Groups Geom. Dyn. 6 (2012), pp. 441–483].

We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong’s hyperbolicity criterion (Ph.D. Thesis) for Coxeter groups built on Danciger–Guéritaud–Kassel’s 2017 work and find examples of Coxeter groups $W$ such that the space of strictly GHC-regular representations of $W$ into $\mathrm {PO}_{d,2}(\mathbb {R})$ up to conjugation is disconnected.

References

Thierry Barbot, Causal properties of AdS-isometry groups. I. Causal actions and limit sets, Adv. Theor. Math. Phys. 12 (2008), no. 1, 1–66. MR 2369412
Thierry Barbot, Deformations of Fuchsian AdS representations are quasi-Fuchsian, J. Differential Geom. 101 (2015), no. 1, 1–46. MR 3356068
T. Barbot, F. Bonsante, J. Danciger, W. M. Goldman, F. Guéritaud, F. Kassel, K. Krasnov, J.-M. Schlenker, and A. Zeghib, Some open questions on anti-de Sitter geometry, ArXiv:1205.6103 (2012).
Yves Benoist, Convexes hyperboliques et quasiisométries, Geom. Dedicata 122 (2006), 109–134 (French, with English summary). MR 2295544, DOI 10.1007/s10711-006-9066-z
Marc Bourdon and Bruce Kleiner, Combinatorial modulus, the combinatorial Loewner property, and Coxeter groups, Groups Geom. Dyn. 7 (2013), no. 1, 39–107. MR 3019076, DOI 10.4171/GGD/177
Thierry Barbot and Quentin Mérigot, Anosov AdS representations are quasi-Fuchsian, Groups Geom. Dyn. 6 (2012), no. 3, 441–483. MR 2961282, DOI 10.4171/GGD/163
N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1349, Hermann, Paris, 1972. MR 0573068
Ruth Charney and Michael Davis, When is a Coxeter system determined by its Coxeter group?, J. London Math. Soc. (2) 61 (2000), no. 2, 441–461. MR 1760693, DOI 10.1112/S0024610799008583
Richard Chow, Groups quasi-isometric to complex hyperbolic space, Trans. Amer. Math. Soc. 348 (1996), no. 5, 1757–1769. MR 1329530, DOI 10.1090/S0002-9947-96-01522-X
Andrew Casson and Douglas Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994), no. 3, 441–456. MR 1296353, DOI 10.1007/BF01231540
D. Cooper, D. D. Long, and A. W. Reid, Infinite Coxeter groups are virtually indicable, Proc. Edinburgh Math. Soc. (2) 41 (1998), no. 2, 303–313. MR 1626421, DOI 10.1017/S0013091500019660
Michael W. Davis, The cohomology of a Coxeter group with group ring coefficients, Duke Math. J. 91 (1998), no. 2, 297–314. MR 1600586, DOI 10.1215/S0012-7094-98-09113-X
Michael W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, NJ, 2008. MR 2360474
Michael W. Davis, The geometry and topology of Coxeter groups, Introduction to modern mathematics, Adv. Lect. Math. (ALM), vol. 33, Int. Press, Somerville, MA, 2015, pp. 129–142. MR 3445448
Yves de Cornulier and Pierre de la Harpe, Décompositions de groupes par produit direct et groupes de Coxeter, Geometric group theory, Trends Math., Birkhäuser, Basel, 2007, pp. 75–102 (French, with English and French summaries). MR 2395791, DOI 10.1007/978-3-7643-8412-8_{7}
Jeffrey Danciger, François Guéritaud, and Fanny Kassel, Convex cocompactness in pseudo-riemannian hyperbolic spaces, Geometriae Dedicata (2017).
Michael W. Davis and Tadeusz Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991), no. 2, 347–388. MR 1131435
Cornelia Druţu and Misha Kapovich, Geometric group theory, A book to appear in the AMS series “Colloquium Publications” (2017).
Alex Eskin and Benson Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997), no. 3, 653–692. MR 1434399, DOI 10.1090/S0894-0347-97-00238-5
Frank Esselmann, The classification of compact hyperbolic Coxeter $d$-polytopes with $d+2$ facets, Comment. Math. Helv. 71 (1996), no. 2, 229–242. MR 1396674, DOI 10.1007/BF02566418
Benson Farb, The quasi-isometry classification of lattices in semisimple Lie groups, Math. Res. Lett. 4 (1997), no. 5, 705–717. MR 1484701, DOI 10.4310/MRL.1997.v4.n5.a8
David Gabai, Convergence groups are Fuchsian groups, Ann. of Math. (2) 136 (1992), no. 3, 447–510. MR 1189862, DOI 10.2307/2946597
Olivier Glorieux and Daniel Monclair, Critical exponent and Hausdorff dimension in pseudo-Riemannian hyperbolic geometry, ArXiv:1606.05512 (2016).
Constantin Gonciulea, Infinite Coxeter groups virtually surject onto $\textbf {Z}$, Comment. Math. Helv. 72 (1997), no. 2, 257–265. MR 1470091, DOI 10.1007/s000140050015
Olivier Guichard and Anna Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012), no. 2, 357–438. MR 2981818, DOI 10.1007/s00222-012-0382-7
Tadeusz Januszkiewicz and Jacek Świa̧tkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003), no. 3, 555–583. MR 1998394, DOI 10.1007/s00014-003-0763-z
Bruce Kleiner and Bernhard Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997), 115–197 (1998). MR 1608566
Bruce Kleiner and Bernhard Leeb, Groups quasi-isometric to symmetric spaces, Comm. Anal. Geom. 9 (2001), no. 2, 239–260. MR 1846203, DOI 10.4310/CAG.2001.v9.n2.a1
François Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), no. 1, 51–114. MR 2221137, DOI 10.1007/s00222-005-0487-3
G. A. Margulis, Factor groups of discrete subgroups and measure theory, Funktsional. Anal. i Prilozhen. 12 (1978), no. 4, 64–76 (Russian). MR 515630
Geoffrey Mess, Lorentz spacetimes of constant curvature, Geom. Dedicata 126 (2007), 3–45. MR 2328921, DOI 10.1007/s10711-007-9155-7
Gabor Moussong, Hyperbolic Coxeter groups, ProQuest LLC, Ann Arbor, MI, 1988. Thesis (Ph.D.)–The Ohio State University. MR 2636665
G. A. Margulis and È. B. Vinberg, Some linear groups virtually having a free quotient, J. Lie Theory 10 (2000), no. 1, 171–180. MR 1748082
Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
Luis Paris, Irreducible Coxeter groups, Internat. J. Algebra Comput. 17 (2007), no. 3, 427–447. MR 2333366, DOI 10.1142/S0218196707003779
Dongwen Qi, On irreducible, infinite, nonaffine Coxeter groups, Fund. Math. 193 (2007), no. 1, 79–93. MR 2284573, DOI 10.4064/fm193-1-5
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234
David A. Stone, Geodesics in piecewise linear manifolds, Trans. Amer. Math. Soc. 215 (1976), 1–44. MR 402648, DOI 10.1090/S0002-9947-1976-0402648-8
Pekka Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346. MR 861709, DOI 10.1007/BF02796595
Pekka Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988), 1–54. MR 961162, DOI 10.1515/crll.1988.391.1
Pavel Tumarkin, Compact hyperbolic Coxeter $n$-polytopes with $n+3$ facets, Electron. J. Combin. 14 (2007), no. 1, Research Paper 69, 36. MR 2350459
È. B. Vinberg, Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1072–1112 (Russian). MR 0302779
È. B. Vinberg, Absence of crystallographic groups of reflections in Lobachevskiĭ spaces of large dimension, Trudy Moskov. Mat. Obshch. 47 (1984), 68–102, 246 (Russian). MR 774946