Geometry & Topology
- Geom. Topol.
- Volume 23, Number 3 (2019), 1251-1337.
The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety
Daniele Alessandrini and Brian Collier
Full-text: Access denied (no subscription detected)
However, an active subscription may be available with MSP at msp.org/gt.
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Abstract
We describe the space of maximal components of the character variety of surface group representations into and .
For every real rank Lie group of Hermitian type, we construct a mapping class group invariant complex structure on the maximal components. For the groups and , we give a mapping class group invariant parametrization of each maximal component as an explicit holomorphic fiber bundle over Teichmüller space. Special attention is put on the connected components which are singular: we give a precise local description of the singularities and their geometric interpretation. We also describe the quotient of the maximal components of and by the action of the mapping class group as a holomorphic submersion over the moduli space of curves.
These results are proven in two steps: first we use Higgs bundles to give a nonmapping class group equivariant parametrization, then we prove an analog of Labourie’s conjecture for maximal –representations.
Article information
Source
Geom. Topol., Volume 23, Number 3 (2019), 1251-1337.
Dates
Received: 27 August 2017
Accepted: 21 July 2018
First available in Project Euclid: 5 June 2019
Permanent link to this document
https://projecteuclid.org/euclid.gt/1559700274
Digital Object Identifier
doi:10.2140/gt.2019.23.1251
Mathematical Reviews number (MathSciNet)
MR3956893
Zentralblatt MATH identifier
07079059
Subjects
Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20]
Secondary: 14H60: Vector bundles on curves and their moduli [See also 14D20, 14F05] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
Keywords
character varieties mapping class group Higgs bundles maximal representations
Citation
Alessandrini, Daniele; Collier, Brian. The geometry of maximal components of the $\mathsf{PSp}(4,\mathbb R)$ character variety. Geom. Topol. 23 (2019), no. 3, 1251--1337. doi:10.2140/gt.2019.23.1251. https://projecteuclid.org/euclid.gt/1559700274
References
- A A'Campo-Neuen, N A'Campo, L Ji, A Papadopoulos, A commentary on Teichmüller's paper “Veränderliche Riemannsche Flächen”, from “Handbook of Teichmüller theory, IV” (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc., Zürich (2014) 805–814 Mathematical Reviews (MathSciNet): MR3289717
- M F Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. 4 (1971) 47–62
- S B Bradlow, O García-Prada, P B Gothen, Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces, Geom. Dedicata 122 (2006) 185–213 Mathematical Reviews (MathSciNet): MR2295550
Zentralblatt MATH: 1132.14029
Digital Object Identifier: doi:10.1007/s10711-007-9127-y - S B Bradlow, O García-Prada, P B Gothen, Deformations of maximal representations in ${\rm Sp}(4,\mathbb R)$, Q. J. Math. 63 (2012) 795–843
- M Burger, A Iozzi, F Labourie, A Wienhard, Maximal representations of surface groups: symplectic Anosov structures, Pure Appl. Math. Q. 1 (2005) 543–590 Mathematical Reviews (MathSciNet): MR2201327
Zentralblatt MATH: 1157.53025
Digital Object Identifier: doi:10.4310/PAMQ.2005.v1.n3.a5 - M Burger, A Iozzi, A Wienhard, Surface group representations with maximal Toledo invariant, Ann. of Math. 172 (2010) 517–566 Mathematical Reviews (MathSciNet): MR2680425
Zentralblatt MATH: 1208.32014
Digital Object Identifier: doi:10.4007/annals.2010.172.517 - M Burger, A Iozzi, A Wienhard, Higher Teichmüller spaces: from ${\rm SL}(2,\mathbb R)$ to other Lie groups, from “Handbook of Teichmüller theory, IV” (A Papadopoulos, editor), IRMA Lect. Math. Theor. Phys. 19, Eur. Math. Soc., Zürich (2014) 539–618 Mathematical Reviews (MathSciNet): MR3289711
- F E Burstall, J H Rawnsley, Twistor theory for Riemannian symmetric spaces: with applications to harmonic maps of Riemann surfaces, Lecture Notes in Mathematics 1424, Springer (1990)
- B Collier, Maximal ${\rm Sp}(4,\mathbb{R})$ surface group representations, minimal immersions and cyclic surfaces, Geom. Dedicata 180 (2016) 241–285 Mathematical Reviews (MathSciNet): MR3451467
Digital Object Identifier: doi:10.1007/s10711-015-0101-9 - B Collier, N Tholozan, J Toulisse, The geometry of maximal representations of surface groups into $\mathrm{SO}(2,n)$, preprint (2017) arXiv: 1702.08799
- K Corlette, Flat $G$–bundles with canonical metrics, J. Differential Geom. 28 (1988) 361–382 Mathematical Reviews (MathSciNet): MR965220
Digital Object Identifier: doi:10.4310/jdg/1214442469
Project Euclid: euclid.jdg/1214442469 - P Deligne, D Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. 36 (1969) 75–109
- S K Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987) 127–131 Mathematical Reviews (MathSciNet): MR887285
Zentralblatt MATH: 0634.53046
Digital Object Identifier: doi:10.1112/plms/s3-55.1.127 - B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
- O García-Prada, P B Gothen, I Mundet i Riera, The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, preprint (2009) arXiv: 0909.4487
- O García-Prada, P B Gothen, I Mundet i Riera, Higgs bundles and surface group representations in the real symplectic group, J. Topol. 6 (2013) 64–118 Mathematical Reviews (MathSciNet): MR3029422
Zentralblatt MATH: 1303.14043
Digital Object Identifier: doi:10.1112/jtopol/jts030 - W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200–225 Mathematical Reviews (MathSciNet): MR762512
Zentralblatt MATH: 0574.32032
Digital Object Identifier: doi:10.1016/0001-8708(84)90040-9 - W M Goldman, Topological components of spaces of representations, Invent. Math. 93 (1988) 557–607 Mathematical Reviews (MathSciNet): MR952283
Zentralblatt MATH: 0655.57019
Digital Object Identifier: doi:10.1007/BF01410200 - W M Goldman, Mapping class group dynamics on surface group representations, from “Problems on mapping class groups and related topics” (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc., Providence, RI (2006) 189–214 Mathematical Reviews (MathSciNet): MR2264541
- P B Gothen, Components of spaces of representations and stable triples, Topology 40 (2001) 823–850 Mathematical Reviews (MathSciNet): MR1851565
Zentralblatt MATH: 1066.14012
Digital Object Identifier: doi:10.1016/S0040-9383(99)00086-5 - P B Gothen, A G Oliveira, Rank two quadratic pairs and surface group representations, Geom. Dedicata 161 (2012) 335–375 Mathematical Reviews (MathSciNet): MR2994046
Zentralblatt MATH: 1256.14033
Digital Object Identifier: doi:10.1007/s10711-012-9709-1 - A Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119–221
- F Guéritaud, O Guichard, F Kassel, A Wienhard, Anosov representations and proper actions, Geom. Topol. 21 (2017) 485–584 Mathematical Reviews (MathSciNet): MR3608719
Zentralblatt MATH: 1373.37095
Digital Object Identifier: doi:10.2140/gt.2017.21.485
Project Euclid: euclid.gt/1510859139 - O Guichard, A Wienhard, Topological invariants of Anosov representations, J. Topol. 3 (2010) 578–642 Mathematical Reviews (MathSciNet): MR2684514
Zentralblatt MATH: 1225.57012
Digital Object Identifier: doi:10.1112/jtopol/jtq018 - O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357–438 Mathematical Reviews (MathSciNet): MR2981818
Zentralblatt MATH: 1270.20049
Digital Object Identifier: doi:10.1007/s00222-012-0382-7 - O Hamlet, Tight maps and holomorphicity, Transform. Groups 19 (2014) 999–1026 Mathematical Reviews (MathSciNet): MR3278859
Digital Object Identifier: doi:10.1007/s00031-014-9283-8 - J L Harer, Stability of the homology of the moduli spaces of Riemann surfaces with spin structure, Math. Ann. 287 (1990) 323–334 Mathematical Reviews (MathSciNet): MR1054572
Zentralblatt MATH: 0715.57004
Digital Object Identifier: doi:10.1007/BF01446896 - N J Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126 Mathematical Reviews (MathSciNet): MR887284
Zentralblatt MATH: 0634.53045
Digital Object Identifier: doi:10.1112/plms/s3-55.1.59 - N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449–473 Mathematical Reviews (MathSciNet): MR1174252
Digital Object Identifier: doi:10.1016/0040-9383(92)90044-I - Z Huang, B Wang, Counting minimal surfaces in quasi-Fuchsian three-manifolds, Trans. Amer. Math. Soc. 367 (2015) 6063–6083 Mathematical Reviews (MathSciNet): MR3356929
Zentralblatt MATH: 1322.53013
Digital Object Identifier: doi:10.1090/tran/6172 - M Kapovich, B Leeb, J Porti, Dynamics on flag manifolds: domains of proper discontinuity and cocompactness, Geom. Topol. 22 (2018) 157–234 Mathematical Reviews (MathSciNet): MR3720343
Zentralblatt MATH: 1381.53090
Digital Object Identifier: doi:10.2140/gt.2018.22.157
Project Euclid: euclid.gt/1513774912 - F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51–114 Mathematical Reviews (MathSciNet): MR2221137
Zentralblatt MATH: 1103.32007
Digital Object Identifier: doi:10.1007/s00222-005-0487-3 - F Labourie, Flat projective structures on surfaces and cubic holomorphic differentials, Pure Appl. Math. Q. 3 (2007) 1057–1099 Mathematical Reviews (MathSciNet): MR2402597
Digital Object Identifier: doi:10.4310/PAMQ.2007.v3.n4.a10 - F Labourie, Cross ratios, Anosov representations and the energy functional on Teichmüller space, Ann. Sci. Éc. Norm. Supér. 41 (2008) 437–469
- F Labourie, Cyclic surfaces and Hitchin components in rank $2$, Ann. of Math. 185 (2017) 1–58 Mathematical Reviews (MathSciNet): MR3583351
Digital Object Identifier: doi:10.4007/annals.2017.185.1.1 - S Lang, On quasi algebraic closure, Ann. of Math. 55 (1952) 373–390 Mathematical Reviews (MathSciNet): MR0046388
Zentralblatt MATH: 0046.26202
Digital Object Identifier: doi:10.2307/1969785 - J C Loftin, Affine spheres and convex $\mathbb{RP}^n$–manifolds, Amer. J. Math. 123 (2001) 255–274
- I G Macdonald, Symmetric products of an algebraic curve, Topology 1 (1962) 319–343 Mathematical Reviews (MathSciNet): MR0151460
Digital Object Identifier: doi:10.1016/0040-9383(62)90019-8 - J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
- D Mumford, Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. 4 (1971) 181–192
- M S Narasimhan, C S Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965) 540–567 Mathematical Reviews (MathSciNet): MR0184252
Zentralblatt MATH: 0171.04803
Digital Object Identifier: doi:10.2307/1970710 - A Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975) 129–152 Mathematical Reviews (MathSciNet): MR0369747
Zentralblatt MATH: 0284.32019
Digital Object Identifier: doi:10.1007/BF01343949 - J Sacks, K Uhlenbeck, Minimal immersions of closed Riemann surfaces, Trans. Amer. Math. Soc. 271 (1982) 639–652 Mathematical Reviews (MathSciNet): MR654854
Zentralblatt MATH: 0527.58008
Digital Object Identifier: doi:10.1090/S0002-9947-1982-0654854-8 - A Schmitt, Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005) 15–49 Mathematical Reviews (MathSciNet): MR2120597
Zentralblatt MATH: 1076.14019
Digital Object Identifier: doi:10.1007/BF02829837 - R M Schoen, The role of harmonic mappings in rigidity and deformation problems, from “Complex geometry” (G Komatsu, Y Sakane, editors), Lecture Notes in Pure and Appl. Math. 143, Dekker, New York (1993) 179–200 Mathematical Reviews (MathSciNet): MR1201611
- R Schoen, S T Yau, Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. of Math. 110 (1979) 127–142 Mathematical Reviews (MathSciNet): MR541332
Zentralblatt MATH: 0431.53051
Digital Object Identifier: doi:10.2307/1971247 - C T Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988) 867–918 Mathematical Reviews (MathSciNet): MR944577
Zentralblatt MATH: 0669.58008
Digital Object Identifier: doi:10.1090/S0894-0347-1988-0944577-9 - C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Inst. Hautes Études Sci. Publ. Math. 79 (1994) 47–129
- C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, II, Inst. Hautes Études Sci. Publ. Math. 80 (1994) 5–79
- A Wienhard, The action of the mapping class group on maximal representations, Geom. Dedicata 120 (2006) 179–191 Mathematical Reviews (MathSciNet): MR2252900
Zentralblatt MATH: 1175.32007
Digital Object Identifier: doi:10.1007/s10711-006-9079-7 - M Wolf, The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989) 449–479 Mathematical Reviews (MathSciNet): MR982185
Digital Object Identifier: doi:10.4310/jdg/1214442885
Project Euclid: euclid.jdg/1214442885
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- Surface Group Representations and U(p, q)-Higgs Bundles
Bradlow, Steven B., García-Prada, Oscar, and Gothen, Peter B., Journal of Differential Geometry, 2003 - Higgs bundles for the Lorentz group
Aparicio Arroyo, Marta and García-Prada, Oscar, Illinois Journal of Mathematics, 2011 - Nonabelianization of Higgs bundles
Hitchin, Nigel and Schaposnik, Laura P., Journal of Differential Geometry, 2014
- Surface Group Representations and U(p, q)-Higgs Bundles
Bradlow, Steven B., García-Prada, Oscar, and Gothen, Peter B., Journal of Differential Geometry, 2003 - Higgs bundles for the Lorentz group
Aparicio Arroyo, Marta and García-Prada, Oscar, Illinois Journal of Mathematics, 2011 - Nonabelianization of Higgs bundles
Hitchin, Nigel and Schaposnik, Laura P., Journal of Differential Geometry, 2014 - The Penrose transform for Sp(n,R) and singular unitary representations
SEKIGUCHI, Hideko, Journal of the Mathematical Society of Japan, 2002 - Composantes de Hitchin et représentations hyperconvexes de
groupes de surface
Guichard, Olivier, Journal of Differential Geometry, 2008 - On type-preserving representations of the four-punctured sphere group
Yang, Tian, Geometry & Topology, 2016 - Higgs bundles for the non-compact dual of the unitary group
García-Prada, Oscar and Oliveira, André G., Illinois Journal of Mathematics, 2011 - The Picard group of the moduli space of curves with level structures
Putman, Andrew, Duke Mathematical Journal, 2012 - Higgs bundles over elliptic curves
Franco, Emilio, Garcia-Prada, Oscar, and Newstead, P. E., Illinois Journal of Mathematics, 2014 - Tropicalization of group representations
Alessandrini, Daniele, Algebraic & Geometric Topology, 2008