Nearly geodesic immersions and domains of discontinuity

Davalo, Colin¹

¹Mathematisches Institut, Ruprecht-Karls Universität Heidelberg, Heidelberg, Germany

Geometry & topology, Tome 29 (2025) no. 5, pp. 2391-2461

Résumé

We study nearly geodesic immersions in higher-rank symmetric spaces of noncompact type, which we define as immersions that satisfy a bound on their fundamental form, generalizing the notion of immersions in hyperbolic space with principal curvature in (−1,1). This notion depends on the choice of a flag manifold embedded in the visual boundary, and immersions satisfying this bound admit a natural domain in this flag manifold that comes with a fibration. As an application we give an explicit fibration of some domains of discontinuity for some Anosov representations. Our method can be applied in particular to some Θ-positive representations for each notion of Θ-positivity.

Détail

DOI : 10.2140/gt.2025.29.2391

Keywords: domains of discontinuity, Anosov representations, symmetric spaces of noncompact type

Bibliographie

[1] D Alessandrini, Higgs bundles and geometric structures on manifolds, Symmetry Integrability Geom. Methods Appl. 15 (2019) 039 | DOI

[2] D Alessandrini, C Davalo, Q Li, Projective structures with (quasi-)Hitchin holonomy, J. Lond. Math. Soc. 110 (2024) | DOI

[3] D Alessandrini, S Maloni, N Tholozan, A Wienhard, Fiber bundles associated with Anosov representations, Forum Math. Sigma 13 (2025) | DOI

[4] Y Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7 (1997) 1 | DOI

[5] J Beyrer, M B Pozzetti, Positive surface group representations in PO(p,q), J. Eur. Math. Soc. (2024) | DOI

[6] J Bochi, R Potrie, A Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. 21 (2019) 3343 | DOI

[7] S Bradlow, B Collier, O García-Prada, P Gothen, A Oliveira, A general Cayley correspondence and higher rank Teichmüller spaces, Ann. of Math. 200 (2024) 803 | DOI

[8] S B Bradlow, O García-Prada, P B Gothen, Deformations of maximal representations in Sp(4, R), Q. J. Math. 63 (2012) 795 | DOI

[9] S Bronstein, Almost-Fuchsian structures on disk bundles over a surface, preprint (2023)

[10] M Burger, A Iozzi, F Labourie, A Wienhard, Maximal representations of surface groups : symplectic Anosov structures, Pure Appl. Math. Q. 1 (2005) 543 | DOI

[11] M Burger, A Iozzi, A Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. 172 (2010) 517 | DOI

[12] M Burger, M B Pozzetti, Maximal representations, non-Archimedean Siegel spaces, and buildings, Geom. Topol. 21 (2017) 3539 | DOI

[13] B Collier, N Tholozan, J Toulisse, The geometry of maximal representations of surface groups into SO0(2,n), Duke Math. J. 168 (2019) 2873 | DOI

[14] D Dumas, A Sanders, Geometry of compact complex manifolds associated to generalized quasi-Fuchsian representations, Geom. Topol. 24 (2020) 1615 | DOI

[15] P B Eberlein, Geometry of nonpositively curved manifolds, Univ. Chicago Press (1996)

[16] C L Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math. 372 (1986) 96 | DOI

[17] C Fevola, Y Mandelshtam, B Sturmfels, Pencils of quadrics: old and new, Matematiche (Catania) 76 (2021) 319 | DOI

[18] V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 | DOI

[19] Ś R Gal, On normal subgroups of Coxeter groups generated by standard parabolic subgroups, Geom. Dedicata 115 (2005) 65 | DOI

[20] M Gromov, Hyperbolic manifolds, groups and actions, from: "Riemann surfaces and related topics", Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 183 | DOI

[21] M Gromov, H B Lawson Jr., W Thurston, Hyperbolic 4–manifolds and conformally flat 3–manifolds, Inst. Hautes Études Sci. Publ. Math. 68 (1988) 27 | DOI

[22] O Guichard, F Labourie, A Wienhard, Positivity and representations of surface groups, preprint (2021)

[23] O Guichard, A Wienhard, Topological invariants of Anosov representations, J. Topol. 3 (2010) 578 | DOI

[24] O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357 | DOI

[25] O Guichard, A Wienhard, Generalizing Lusztig’s total positivity, Invent. Math. 239 (2025) 707 | DOI

[26] S Helgason, Differential geometry, Lie groups, and symmetric spaces, 80, Academic (1978) | DOI

[27] N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449 | DOI

[28] M Kapovich, B Leeb, Finsler bordifications of symmetric and certain locally symmetric spaces, Geom. Topol. 22 (2018) 2533 | DOI

[29] M Kapovich, B Leeb, J Porti, Anosov subgroups: dynamical and geometric characterizations, Eur. J. Math. 3 (2017) 808 | DOI

[30] M Kapovich, B Leeb, J Porti, Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geom. Topol. 22 (2018) 157 | DOI

[31] M Kapovich, B Leeb, J Porti, A Morse lemma for quasigeodesics in symmetric spaces and Euclidean buildings, Geom. Topol. 22 (2018) 3827 | DOI

[32] F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51 | DOI

[33] A L Onishchik, È B Vinberg, Lie groups and algebraic groups, Springer (1990) | DOI

[34] P Planche, Géométrie de Finsler sur les espaces symétriques, PhD thesis, Université de Genève (1995)

[35] J M Riestenberg, A quantified local-to-global principle for Morse quasigeodesics, Groups Geom. Dyn. 19 (2025) 37 | DOI

[36] K K Uhlenbeck, Closed minimal surfaces in hyperbolic 3–manifolds, from: "Seminar on minimal submanifolds", Ann. of Math. Stud. 103, Princeton Univ. Press (1983) 147 | DOI

[37] A Wienhard, An invitation to higher Teichmüller theory, from: "Proceedings of the International Congress of Mathematicians, II", World Sci. (2018) 1013

[38] M Wiggerman, The fundamental group of a real flag manifold, Indag. Math. 9 (1998) 141 | DOI