From hyperbolic Dehn filling to surgeries in representation varieties

In this semi-expository article we describe a gluing method developed for constructing certain model objects in representation varieties Hom (π1 (Σ) , G) for a topological surface Σ and a semisimple Lie group G. Explicit examples are demonstrated in the case of Θ-positive representations lying in the p·(2g − 2)−1 many exceptional connected components of the SO (p, p+ 1)-character variety for p > 2.

[1]  Moduli spaces of Higgs bundles on degenerating Riemann surfaces , 2015, 1507.04382.

[2]  N. Hitchin.  THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE , 1987 .

[3]  Andr'es Sambarino,et al.  The pressure metric for Anosov representations , 2013, 1301.7459.

[4]  M. Arroyo.  The geometry of the so(p,q)-higgs bundles , 2009 .

[5]  W. Thurston.  The geometry and topology of three-manifolds , 1979 .

[6]  M. Lackenby,et al.  The maximal number of exceptional Dehn surgeries , 2008, 0808.1176.

[7]  D. Rolfsen.  Rational surgery calculus: extension of Kirby's theorem. , 1984 .

[8]  Tengren Zhang,et al.  Collar lemma for Hitchin representations , 2014, 1411.2082.

[9]  B. Kostant,et al.  The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group , 1959 .

[10]  Beatrice Pozzetti,et al.  Positive surface group representations in PO(p,q) , 2021, 2106.14725.

[11]  Olivier Y Guichard,et al.  Topological invariants of Anosov representations , 2009, 0907.0273.

[12]  M. Burger,et al.  Surface group representations with maximal Toledo invariant , 2003, math/0605656.

[13]  J. Millson.  On the first Betti number of a constant negatively curved manifold , 1976 .

[14]  M. Raghunathan.  Discrete subgroups of Lie groups , 1972 .

[15]  C. Simpson.  Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization , 1988 .

[16]  A. Bobenko.  Compact Riemann Surfaces , 2005 .

[17]  Olivier Guichard.  Composantes de Hitchin et représentations hyperconvexes de groupes de surface , 2008 .

[18]  Brian Collier.  SO(n,n+1)-surface group representations and their Higgs bundles , 2017, 1710.01287.

[19]  N. Hitchin.  LIE-GROUPS AND TEICHMULLER SPACE , 1992 .

[20]  M. Dehn,et al.  Über die Topologie des dreidimensionalen Raumes , 1910 .

[21]  Anna Wienhard.  AN INVITATION TO HIGHER TEICHMÜLLER THEORY , 2019, Proceedings of the International Congress of Mathematicians (ICM 2018).

[22]  Michael Wolf.  The Teichmüller theory of harmonic maps , 1989 .

[23]  Friedhelm Waldhausen,et al.  On irreducible 3-manifolds which are sufficiently large * , 2010 .

[24]  W. B. R. Lickorish,et al.  A Representation of Orientable Combinatorial 3-Manifolds , 1962 .

[26]  W. Goldman.  GEOMETRIC STRUCTURES ON MANIFOLDS AND VARIETIES OF REPRESENTATIONS , 1988 .

[27]  S. Donaldson.  Geometry of four-manifolds , 1990 .

[28]  Anosov representations: domains of discontinuity and applications , 2011, 1108.0733.

[29]  Beatrice Pozzetti,et al.  A collar lemma for partially hyperconvex surface group representations , 2020, Transactions of the American Mathematical Society.

[30]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[31]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[32]  J. Millson,et al.  Deformation Spaces Associated to Compact Hyperbolic Manifolds , 1986 .

[33]  J. D. Miller.  Modifications and cobounding manifolds , 1967 .

[34]  J. Morgan,et al.  On Thurston''s uniformization theorem for three-dimensional manifolds , 1984 .

[35]  Lorenzo Foscolo.  A gluing construction for periodic monopoles , 2014, 1411.6951.

[36]  H. Konno.  Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface , 1993 .

[37]  Basmajian's identity in higher Teichmüller–Thurston theory , 2015, 1504.00649.

[38]  O. G. Prada.  Higgs bundles and higher Teichmüller spaces , 2020 .

[39]  C. S. Seshadri,et al.  Moduli of vector bundles on curves with parabolic structures , 1980 .

[40]  C. Simpson.  Higgs bundles and local systems , 1992 .

[41]  C. Taubes.  Self-dual Yang-Mills connections on non-self-dual 4-manifolds , 1982 .

[42]  Vorlesungen über die Theorie der automorphen Funktionen , 1914 .

[43]  W. Goldman.  Topological components of spaces of representations , 1988 .

[44]  S. Cappell,et al.  Self-Adjoint Elliptic Operators and Manifold Decompositions Part I: Low Eigenmodes and Stretching , 1996 .

[45]  ON THE CAPPELL-LEE-MILLER GLUING THEOREM , 1998, math/9803154.

[46]  W. Haken.  Theorie der Normalflächen , 1961 .

[47]  A. Reid.  A non-Haken hyperbolic $3$-manifold covered by a surface bundle. , 1995 .

[48]  Hao Sun,et al.  Topological invariants of parabolic G-Higgs bundles , 2018, Mathematische Zeitschrift.

[49]  W. Goldman.  The Symplectic Nature of Fundamental Groups of Surfaces , 1984 .

[50]  Andr'es Sambarino,et al.  An introduction to pressure metrics for higher Teichmüller spaces , 2015, Ergodic Theory and Dynamical Systems.

[51]  M. Bridgeman.  Average bending of convex pleated planes in hyperbolic three-space , 1998 .

[52]  S. Donaldson.  Twisted harmonic maps and the self-duality equations , 1987 .

[53]  W. Haken,et al.  Über das Homöomorphieproblem der 3-Mannigfaltigkeiten. I , 1962 .

[54]  Olivier Y Guichard,et al.  Positivity and higher Teichm\"uller theory , 2018, 1802.02833.

[55]  A. Schmitt.  Geometric Invariant Theory and Decorated Principal Bundles , 2008 .

[56]  Christos Kourouniotis,et al.  Deformations of hyperbolic structures , 1985, Mathematical Proceedings of the Cambridge Philosophical Society.

[57]  J. Luecke.  Dehn Surgery on Knots in the 3-Sphere , 1995 .

[58]  M. B. Pozzetti,et al.  Basmajian-type inequalities for maximal representations , 2016, Journal of Differential Geometry.

[59]  R. Potrie,et al.  Eigenvalues and entropy of a Hitchin representation , 2014, Inventiones mathematicae.

[60]  C. Hodgson,et al.  DEHN FILLING: A SURVEY , 2011 .

[61]  Jürgen Jost,et al.  Compact Riemann Surfaces - An Introduction to Contemporary Mathematics, Third Edition , 2006, Universitext.

[62]  S. Bradlow,et al.  SO(p,q)-Higgs bundles and higher Teichm\"uller components , 2018 .

[63]  Georgios Kydonakis.  Model Higgs Bundles in Exceptional Components of the Sp(4,R)-Character Variety , 2018, JOURNAL OF ADVANCES IN MATHEMATICS.

[64]  F. Labourie.  Cross ratios, surface groups, PSL(n,ℝ) and diffeomorphisms of the circle , 2007 .

[65]  Oscar Garcia-Prada,et al.  Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group , 2015, Advances in Mathematics.

[66]  Siqi He.  A gluing theorem for the Kapustin–Witten equations with a Nahm pole , 2017, Journal of Topology.

[67]  Ulrich Oertel,et al.  An algorithm to decide if a 3-manifold is a Haken manifold , 1984 .

[68]  Universal bounds for hyperbolic Dehn surgery , 2002, math/0204345.

[69]  S. Wolpert.  The Fenchel-Nielsen deformation , 1982 .

[70]  M. Boileau,et al.  Geometrization of 3-orbifolds of cyclic type , 1998, math/9805073.

[71]  Andr'e Oliveira,et al.  A general Cayley correspondence and higher Teichm\"uller spaces , 2021 .

[72]  J. Hubbard.  Teichmüller Theory and Applications to Geometry, Topology, and Dynamics , 2016 .

[73]  Moduli spaces of local systems and higher Teichmüller theory , 2003, math/0311149.

[74]  M. B. Pozzetti,et al.  Maximal representations, non Archimedean Siegel spaces, and buildings , 2015, 1509.01184.

[75]  Carlos Simpson,et al.  Harmonic bundles on noncompact curves , 1990 .

[76]  Gabriele Mondello.  Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy , 2016, 1607.04634.

[77]  The moduli space of hyperbolic cone structures , 1998, math/9805129.

[78]  S. Boyer.  Dehn surgery on knots , 1998 .

[79]  Bending in the space of quasi-Fuchsian structures , 1991 .

[80]  C. Kourouniotis.  Discrete Groups and Geometry: The geometry of bending quasi-Fuchsian groups , 1992 .

[81]  D. Epstein,et al.  Fundamentals of Hyperbolic Manifolds: CONVEX HULLS IN HYPERBOLIC SPACE, A THEOREM OF SULLIVAN, AND MEASURED PLEATED SURFACES , 2006 .

[82]  C. Ehresmann.  Les connexions infinitésimales dans un espace fibré différentiable , 1951 .

[83]  J. Swoboda,et al.  Ends of the moduli space of Higgs bundles , 2014, 1405.5765.

[84]  Moduli for decorated tuples of sheaves and representation spaces for quivers , 2004, math/0401173.

[85]  Kevin Corlette,et al.  Flat $G$-bundles with canonical metrics , 1988 .

[86]  F. Labourie,et al.  Cross ratios and identities for higher Teichmüller-Thurston theory , 2006, math/0611245.

[87]  P. Boalch,et al.  Wild non-abelian Hodge theory on curves , 2001, Compositio Mathematica.

[88]  G. Lusztig.  Total Positivity in Reductive Groups , 2019 .

[89]  W. Goldman.  Flat Affine, Projective and Conformal Structures on Manifolds: A Historical Perspective , 2019, Geometry in History.

[90]  Maximal Representations of Surface Groups: Symplectic Anosov Structures , 2005, math/0506079.

[91]  Gang Tian,et al.  The Geometrization Conjecture , 2014 .

[92]  Yi Huang,et al.  McShane Identities for Higher Teichmüller Theory and the Goncharov–Shen Potential , 2019, Memoirs of the American Mathematical Society.

[93]  Michael Wolf.  Infinite energy harmonic maps and degeneration of hyperbolic surfaces in moduli space , 1991 .

[94]  O. García-Prada,et al.  The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations , 2009, 0909.4487.