Divisible convex sets with properly embedded cones

Pierre-Louis Blayac; Gabriele Viaggi

doi:10.1007/s10240-024-00152-y

Article

Divisible convex sets with properly embedded cones

Pierre-Louis Blayac ¹ ; Gabriele Viaggi ¹

¹

Publications Mathématiques de l'IHÉS, Volume 141 (2025), pp. 99-189

Abstract

In this article we construct many examples of properly convex irreducible domains divided by Zariski dense relatively hyperbolic groups in every dimension at least 3. This answers a question of Benoist. Relative hyperbolicity and non-strict convexity are captured by a family of properly embedded cones (convex hulls of points and ellipsoids) in the domain. Our construction is most flexible in dimension 3 where we give a purely topological criterion for the existence of a large deformation space of geometrically controlled convex projective structures with totally geodesic boundary on a compact 3-manifold.

Received: 2023-03-09
Accepted: 2024-09-16
Online First: 2024-10-01
Published online: 2025-06-07

Zbl

DOI: 10.1007/s10240-024-00152-y

Author's affiliations:

Pierre-Louis Blayac ¹; Gabriele Viaggi ¹

¹

@article{PMIHES_2025__141__99_0,
     author = {Pierre-Louis Blayac and Gabriele Viaggi},
     title = {Divisible convex sets with properly embedded cones},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {99--189},
     year = {2025},
     publisher = {Springer International Publishing},
     address = {Cham},
     volume = {141},
     doi = {10.1007/s10240-024-00152-y},
     zbl = {08054049},
     language = {en},
     url = {https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00152-y/}
}

TY  - JOUR
AU  - Pierre-Louis Blayac
AU  - Gabriele Viaggi
TI  - Divisible convex sets with properly embedded cones
JO  - Publications Mathématiques de l'IHÉS
PY  - 2025
SP  - 99
EP  - 189
VL  - 141
PB  - Springer International Publishing
PP  - Cham
UR  - https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00152-y/
DO  - 10.1007/s10240-024-00152-y
LA  - en
ID  - PMIHES_2025__141__99_0
ER  -

%0 Journal Article
%A Pierre-Louis Blayac
%A Gabriele Viaggi
%T Divisible convex sets with properly embedded cones
%J Publications Mathématiques de l'IHÉS
%D 2025
%P 99-189
%V 141
%I Springer International Publishing
%C Cham
%U https://pmihes.centre-mersenne.org/articles/10.1007/s10240-024-00152-y/
%R 10.1007/s10240-024-00152-y
%G en
%F PMIHES_2025__141__99_0

Pierre-Louis Blayac; Gabriele Viaggi. Divisible convex sets with properly embedded cones. Publications Mathématiques de l'IHÉS, Volume 141 (2025), pp. 99-189. doi: 10.1007/s10240-024-00152-y

References
Cited by

[1.] S. A. Ballas; L. Marquis Properly convex bending of hyperbolic manifolds, Groups Geom. Dyn., Volume 14 (2020), pp. 653-688 | MR | Zbl | DOI

[2.] S. A. Ballas; J. Danciger; G.-S. Lee Convex projective structures on nonhyperbolic three-manifolds, Geom. Topol., Volume 22 (2018), pp. 1593-1646 | MR | Zbl | DOI

[3.] S. A. Ballas; D. Cooper; A. Leitner Generalized cusps in real projective manifolds: classification, J. Topol., Volume 13 (2020), pp. 1455-1496 | MR | Zbl | DOI

[4.] W. Ballmann Nonpositively curved manifolds of higher rank, Ann. Math., Volume 122 (1985), pp. 597-609 | MR | Zbl | DOI

[5.] T. Barbot; F. Bonsante; J.-M. Schlenker Collisions of particles in locally AdS spacetimes I. Local description and global examples, Commun. Math. Phys., Volume 308 (2011), pp. 147-200 | MR | DOI | Zbl

[6.] Y. Benoist Automorphismes des cônes convexes, Invent. Math., Volume 141 (2000), pp. 149-193 | MR | DOI | Zbl

[7.] Y. Benoist Convex divisible II, Duke Math. J., Volume 120 (2003), pp. 97-120 | MR | DOI | Zbl

[8.] Y. Benoist Convex Divisible I, in Algebraic Groups and Arithmetic, 17, 2004, pp. 339-374 | Zbl

[9.] Y. Benoist Convex divisible III, Ann. Sci. ENS, Volume 38 (2005), pp. 793-832 | MR | Zbl | Numdam

[10.] Y. Benoist Convex divisible IV, Invent. Math., Volume 164 (2006), pp. 249-278 | MR | Zbl | DOI

[11.] Y. Benoist A survey on divisible convex sets, Geometry, Analysis and Topology of Discrete Groups, 6, Int. Press, Somerville, 2008, pp. 1-18 | Zbl

[12.] Y. Benoist Five lectures on lattices in semisimple Lie groups, Géométries À Courbure Négative Ou Nulle, Groupes Discrets et Rigidité, 18, Soc. Math. France, Paris, 2009, pp. 117-176 | Zbl

[13.] Y. Benoist Exercises on Divisible Convex Sets, 2012 (https://www.imo.universite-paris-saclay.fr/~benoist/prepubli/12GearJuniorRetreat.pdf)

[14.] J.-P. Benzécri Sur les variétés localement affines et localement projectives, Bull. Soc. Math. Fr., Volume 88 (1960), pp. 229-332 | DOI | Zbl | Numdam

[15.] N. Bergeron; F. Haglund; D. T. Wise Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc., Volume 83 (2011), pp. 431-448 | MR | DOI | Zbl

[16.] G. Birkhoff Extensions of Jentzsch’s theorem, Trans. Am. Math. Soc., Volume 85 (1957), pp. 219-227 | MR | Zbl

[17.] P.-L. Blayac, Dynamical aspects of convex projective structures, PhD thesis, Laboratoire Alexander Grothendieck, Université Paris-Saclay, 2021.

[18.] P.-L. Blayac, Patterson-Sullivan densities in convex projective geometry, Comment. Math. Helv., to appear, | arXiv

[19.] P.-L. Blayac; F. Zhu Ergodicity an equidistribution in Hilbert geometry, J. Mod. Dyn., Volume 19 (2023), pp. 879-945 | MR | Zbl | DOI

[20.] M. D. Bobb Convex projective manifolds with a cusp of any non-diagonalizable type, J. Lond. Math. Soc. (2), Volume 100 (2019), pp. 183-202 | MR | DOI | Zbl

[21.] F. Bonahon; J.-P. Otal Laminations mesurées de plissage des variétés hyperboliques de dimension 3, Ann. Math., Volume 160 (2004), pp. 1013-1055 | MR | DOI | Zbl

[22.] A. Borel Compact Clifford-Klein forms of symmetric spaces, Topology, Volume 2 (1963), pp. 111-122 | MR | DOI | Zbl

[23.] B. Bowditch Relatively hyperbolic groups, Int. J. Algebra Comput., Volume 22 (2012), pp. 1-66 | MR | Zbl | DOI

[24.] K. Burns; R. Spatzier Manifolds of nonpositive curvature and their buildings, Publ. Math. IHES, Volume 65 (1987), pp. 35-59 | MR | Zbl | Numdam | DOI

[25.] R. Canary, Anosov representations: Informal lecture notes, preprint, available at http://www.math.lsa.umich.edu/~canary/lecnotespublic.pdf.

[26.] S. Choi; W. M. Goldman Convex real projective structures on closed surfaces are closed, Proc. Am. Math. Soc., Volume 118 (1993), pp. 657-661 | MR | Zbl | DOI

[27.] Y.-E. Choi; C. Series Lengths are coordinates for convex structures, J. Differ. Geom., Volume 73 (2006), pp. 75-116 | MR | Zbl | DOI

[28.] S. Choi; G.-S. Lee; L. Marquis Convex projective generalized Dehn filling, Ann. Sci. Éc. Norm. Supér., Volume 53 (2020), pp. 217-266 | MR | Zbl | DOI

[29.] J. Danciger A geometric transition from hyperbolic to anti-de Sitter geometry, Geom. Topol., Volume 17 (2013), pp. 3077-3134 | MR | DOI | Zbl

[30.] J. Danciger, F. Guéritaud and F. Kassel, Convex-cocompact actions in real projective geometry, Ann. Sci. Éc. Norm. Supér., to appear, | arXiv

[31.] J. Danciger, F. Guéritaud, F. Kassel, G.-S. Lee and L. Marquis, Convex compactness for Coxeter groups, J. Eur. Math. Soc., to appear, | arXiv

[32.] P. de la Harpe On Hilbert’s metric for simplices (G. A. Niblo; M. A. Roller, eds.), Geometric Group Theory, Cambridge University Press, Cambridge, 1993, pp. 97-119 | DOI | Zbl

[33.] V. Gadre The limit set of the handlebody set has measure zero, appendix to Are large distance Heegaard splittings generic? by M. Lustig and Y. Moriah, J. Reine Angew. Math., Volume 670 (2012), pp. 117-119 | MR

[34.] M. Gromov; W. Thurston Pinching constants for hyperbolic manifolds, Invent. Math., Volume 89 (1987), pp. 1-12 | MR | Zbl | DOI

[35.] M. Islam, Rank-one Hilbert geometries, Geom. Topol., to appear, | arXiv

[36.] M. Islam; A. Zimmer Convex co-compact actions of relatively hyperbolic groups, Geom. Topol., Volume 29 (2023), pp. 470-511 | Zbl

[37.] M. Islam and A. Zimmer, The structure of relatively hyperbolic groups in convex real projective geometry, preprint, | arXiv

[38.] D. Johnson; J. Millson Deformation Spaces Associated to Compact Hyperbolic Manifolds, Birkhäuser, Basel, 1986 | DOI

[39.] V. Kac; E. Vinberg Quasihomogeneous cones, Math. Notes, Volume 1 (1967), pp. 231-235 | DOI | Zbl

[40.] M. Kapovich Convex projective structures on Gromov-Thurston manifolds, Geom. Topol., Volume 11 (2007), pp. 1777-1830 | MR | Zbl | DOI

[41.] M. Kapovich Hyperbolic Manifolds and Discrete Groups, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, 2009

[42.] S. Kerckhoff The measure of the limit set of the handlebody group, Topology, Volume 29 (1990), pp. 27-40 | MR | Zbl | DOI

[43.] M. Koecher The Minnesota Notes on Jordan Algebras and Their Applications, 1710, Springer, Berlin, 1999 (Edited, annotated and with a preface by Aloys Krieg and Sebastian Walcher) | Zbl | DOI

[44.] J.-L. Koszul Déformation des connexions localement plates, Ann. Inst. Fourier, Volume 18 (1968), pp. 103-114 | MR | DOI | Zbl | Numdam

[45.] C. Lecuire An extension of the Masur domain (Y. Minsky; M. Sakuma; C. Series, eds.), Spaces of Kleinian Groups, Cambridge University Press, Cambridge, 2019, pp. 49-74

[46.] G.-S. Lee; L. Marquis; S. Riolo A small closed convex projective 4-manifold via Dehn filling, Publ. Mat., Volume 66 (2022), pp. 369-403 | MR | DOI | Zbl

[47.] L. Marquis Espace des modules de certains polyèdres projectifs miroirs, Geom. Dedic., Volume 147 (2010), pp. 47-86 | Zbl | DOI

[48.] L. Marquis Exemples de variétés projectives strictement convexes de volume fini en dimension quelconque, Enseign. Math., Volume 58 (2012), pp. 3-47 | MR | Zbl | DOI

[49.] L. Marquis, Around groups in Hilbert geometry, in G. Besson, M. Troyanov and A. Papadopoulos (eds.) Handbook of Hilbert geometry.

[50.] B. Martelli An Introduction to Geometric Topology, 2016 (CreateSpace Independent Publishing Platform) | Zbl

[51.] J. S. Milne, Algebraic Number Theory, https://jmilne.org/math/CourseNotes/ANT.pdf.

[52.] B. Nica, Linear groups - Malcev’s theorem and Selberg’s lemma, preprint, | arXiv

[53.] D. Osin Acylindrically hyperbolic groups, Trans. Am. Math. Soc., Volume 368 (2016), pp. 851-888 | MR | Zbl | DOI

[54.] G. Prasad; R. Rapinchuck Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. IHES, Volume 109 (2009), pp. 113-184 | MR | DOI | Zbl | Numdam

[55.] S. Riolo; A. Seppi Geometric transition from hyperbolic to anti-de Sitter structures in dimension four, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 23 (2022), pp. 115-176 | MR | Zbl

[56.] V. Schroeder Analytic manifolds of nonpositive curvature with higher rank subspaces, Arch. Math., Volume 56 (1991), pp. 81-85 | MR | Zbl | DOI

[57.] P. Scott; T. Wall Topological Methods in Group Theory, in Homological Group Theory, 36 (1979), pp. 137-203 | DOI

[58.] J.-P. Serre Arbres, Amalgames, ${SL}_{2}$ , Rédigé Avec la Collaboration de Hyman Bass, 46, Société Mathématique de France, Paris, 1977 | Zbl | Numdam

[59.] J.-P. Serre Cohomologie des groupes discrets, Prospects in Mathematics, (AM-70), 70, Princeton University Press, Princeton, 2016, pp. 77-170

[60.] C. Siegel Indefinite quadratische formen und funktionentheorie, Math. Ann., Volume 124 (1951), pp. 17-54 (364–387) | MR | Zbl | DOI

[61.] W. Thurston Geometry and Topology of Three-Manifolds, 1979 (Princeton lecture notes)

[62.] W. Thurston Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. (N.S.), Volume 6 (1982), pp. 357-381 | MR | DOI | Zbl

[63.] W. Thurston, Hyperbolic structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary, preprint, | arXiv

[64.] P. Tukia Conical limit points and uniform convergence groups, J. Reine Angew. Math., Volume 501 (1998), pp. 71-98 | MR | Zbl | DOI

[65.] J. Vey Sur les automorphismes affines des ouverts convexes saillants, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 24 (1970), pp. 641-665 | MR | Zbl | Numdam

[66.] E. Vinberg The theory of homogeneous convex cones, Trudy Moskov. Mat. Obsc., Volume 12 (1963), pp. 303-358 | MR | Zbl

[67.] E. Vinberg Structure of the group of automorphisms of a homogeneous convex cone, Trudy Moskov. Mat. Obsc., Volume 13 (1965), pp. 56-83 | MR | Zbl

[68.] E. Vinberg Discrete linear groups that are generated by reflections, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 35 (1971), pp. 1072-1112 | MR | Zbl

[69.] E. Vinberg Rings of definition of dense subgroups of semisimple linear groups, Math. USSR, Izv., Volume 5 (1971), pp. 45-55 | MR | Zbl | DOI

[70.] T. Weisman Dynamical properties of convex-cocompact actions in projective space, J. Topol., Volume 16 (2023), pp. 990-1047 | MR | DOI | Zbl

[71.] T. Weisman, An extended definition of Anosov representation for relatively hyperbolic groups, preprint, | arXiv

[72.] T. Weisman, Examples of extended geometrically finite representations, preprint, | arXiv

[73.] D. Witte Morris An Introduction to Arithmetic Groups, 2015 (Deductive Press) | Zbl

[74.] A. Yaman A topological characterization of relative hyperbolic groups, J. Reine Angew. Math., Volume 566 (2004), pp. 41-89 | MR | Zbl

[75.] A. Zimmer A higher rank rigidity theorem for convex real projective manifolds, Geom. Topol., Volume 27 (2023), pp. 2899-2936 | MR | DOI | Zbl

Cited by Sources: