Maximal representations, non-Archimedean Siegel spaces, and buildings

Burger, Marc¹ ; Pozzetti, Maria Beatrice²

¹ Department of Mathematics, ETH Zentrum, Zürich, Switzerland
² Mathematics Institute, University of Warwick, Coventry, United Kingdom

Geometry & topology, Tome 21 (2017) no. 6, pp. 3539-3599.

Résumé

Let $\mathbb{F}$ be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in $Sp\left(2n,\mathbb{F}\right)$. We show that ultralimits of maximal representations in $Sp\left(2n,ℝ\right)$ admit such a framing, and that all maximal framed representations satisfy a suitable generalization of the classical collar lemma. In particular, this establishes a collar lemma for all maximal representations into $Sp\left(2n,ℝ\right)$. We then describe a procedure to get from representations in $Sp\left(2n,\mathbb{F}\right)$ interesting actions on affine buildings, and in the case of representations admitting a maximal framing, we describe the structure of the elements of the group acting with zero translation length.

Détail

DOI : 10.2140/gt.2017.21.3539

Classification : 20-XX, 22E40
Keywords: maximal representation, affine building, collar lemma, real closed field, non-Archimedean symmetric spaces

Bibliographie

[1] D Alessandrini, Tropicalization of group representations, Algebr. Geom. Topol. 8 (2008) 279 | DOI

[2] F Bruhat, J Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972) 5

[3] G W Brumfiel, The real spectrum compactification of Teichmüller space, from: "Geometry of group representations" (editors W M Goldman, A R Magid), Contemp. Math. 74, Amer. Math. Soc. (1988) 51 | DOI

[4] G W Brumfiel, The tree of a non-Archimedean hyperbolic plane, from: "Geometry of group representations" (editors W M Goldman, A R Magid), Contemp. Math. 74, Amer. Math. Soc. (1988) 83 | DOI

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

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

[7] B Collier, Q Li, Asymptotics of Higgs bundles in the Hitchin component, Adv. Math. 307 (2017) 488 | DOI

[8] P Erdös, L Gillman, M Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. 61 (1955) 542 | DOI

[9] I Kaplansky, Linear algebra and geometry: a second course, Dover (2003)

[10] L Katzarkov, A Noll, P Pandit, C Simpson, Constructing buildings and harmonic maps, preprint (2015)

[11] B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. 86 (1997) 115

[12] L Kramer, S Shelah, K Tent, S Thomas, Asymptotic cones of finitely presented groups, Adv. Math. 193 (2005) 142 | DOI

[13] L Kramer, K Tent, Asymptotic cones and ultrapowers of Lie groups, Bull. Symbolic Logic 10 (2004) 175 | DOI

[14] I Le, Higher laminations and affine buildings, Geom. Topol. 20 (2016) 1673 | DOI

[15] G S Lee, T Zhang, Collar lemma for Hitchin representations, Geom. Topol. 21 (2017) 2243 | DOI

[16] A H Lightstone, A Robinson, Nonarchimedean fields and asymptotic expansions, North-Holland (1975)

[17] G Lion, M Vergne, The Weil representation, Maslov index and theta series, 6, Birkhäuser (1980)

[18] J Loftin, Convex R‘P2 structures and cubic differentials under neck separation, preprint (2015)

[19] R Mazzeo, J Swoboda, H Weiss, F Witt, Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016) 2227 | DOI

[20] A Parreau, Immeubles affines : construction par les normes et étude des isométries, from: "Crystallographic groups and their generalizations" (editors P Igodt, H Abels, Y Félix, F Grunewald), Contemp. Math. 262, Amer. Math. Soc. (2000) 263 | DOI

[21] A Parreau, Sous-groupes elliptiques de groupes linéaires sur un corps valué, J. Lie Theory 13 (2003) 271

[22] A Parreau, Compactification d’espaces de représentations de groupes de type fini, Math. Z. 272 (2012) 51 | DOI

[23] A Parreau, Invariant subspaces for some surface groups acting on A2–Euclidean buildings, preprint (2015)

[24] A Robinson, Function theory on some nonarchimedean fields, Amer. Math. Monthly 80 (1973) 87 | DOI

[25] I Satake, Algebraic structures of symmetric domains, 4, Iwanami Shoten (1980)

[26] C L Siegel, Symplectic geometry, Amer. J. Math. 65 (1943) 1 | DOI

[27] T Strubel, Fenchel–Nielsen coordinates for maximal representations, Geom. Dedicata 176 (2015) 45 | DOI

[28] B Thornton, Asymptotic cones of symmetric spaces, PhD thesis, The University of Utah (2002)

[29] T Zhang, The degeneration of convex RP2 structures on surfaces, Proc. Lond. Math. Soc. 111 (2015) 967 | DOI

[30] T Zhang, Degeneration of Hitchin representations along internal sequences, Geom. Funct. Anal. 25 (2015) 1588 | DOI