Centre de diffusion de revues académiques mathématiques

 
 
 
 

Séminaire de Théorie spectrale et géométrie (Grenoble)

Table des matières de ce volume | Article précédent | Article suivant
Michael Heusener
Some recent results about the $\mathrm{SL}_n(\mathbb{C})$–representation spaces of knot groups
Séminaire de Théorie spectrale et géométrie (Grenoble), 32 (2014-2015), p. 137-161, doi: 10.5802/tsg.307
Article PDF
Class. Math.: 57M25, 57M05, 57M27
Mots clés: knot group, representation variety, character variety

Résumé - Abstract

This survey reviews some facts about about the representation and character varieties of knot groups into $\mathrm{SL}_n(\mathbb{C})$ with $n\ge 3$ are presented. This concerns mostly joint work of the author with L. Ben Abdelghani, O. Medjerab, V. Muños and J. Porti.

Bibliographie

[1] Michael Artin, “On the solutions of analytic equations”, Invent. Math. 5 (1968), p. 277-291  MR 232018 |  Zbl 0172.05301
[2] Leila Ben Abdelghani, “Espace des représentations du groupe d’un nœud classique dans un groupe de Lie”, Ann. Inst. Fourier (Grenoble) 50 (2000) no. 4, p. 1297-1321  MR 1799747 |  Zbl 0956.57006
[3] Leila Ben Abdelghani, “Tangent cones and local geometry of the representation and character varieties of knot groups”, Algebr. Geom. Topol. 10 (2010) no. 1, p. 433-463 Article |  MR 2602842 |  Zbl 1201.57002
[4] Leila Ben Abdelghani & Michael Heusener, “Irreducible representations of knot groups into SL(n,C)”, to appear in Publicacions Matemàtiques, https://arxiv.org/abs/1111.2828, 2015
[5] Leila Ben Abdelghani, Michael Heusener & Hajer Jebali, “Deformations of metabelian representations of knot groups into ${\rm SL}(3,{\bf C})$”, J. Knot Theory Ramifications 19 (2010) no. 3, p. 385-404 Article |  MR 2646638 |  Zbl 1195.57023
[6] Nicolas Bergeron, Elisha Falbel & Antonin Guilloux, “Tetrahedra of flags, volume and homology of ${\rm SL}(3)$”, Geom. Topol. 18 (2014) no. 4, p. 1911-1971 Article |  MR 3268771
[7] Hans U. Boden & Stefan Friedl, “Metabelian ${\rm SL}(n,\mathbb{C})$ representations of knot groups”, Pacific J. Math. 238 (2008) no. 1, p. 7-25 Article |  MR 2443505 |  Zbl 1154.57004
[8] Hans U. Boden & Stefan Friedl, “Metabelian ${\rm SL}(n,\mathbb{C})$ representations of knot groups, III: deformations”, Q. J. Math. 65 (2014) no. 3, p. 817-840  MR 3261968
[9] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer-Verlag, 1994, Corrected reprint of the 1982 original  MR 1324339 |  Zbl 0584.20036
[10] Michelle Bucher, Marc Burger & Alessandra Iozzi, “Rigidity of representations of hyperbolic lattices ${\Gamma } <\mathrm{PSL}(2,\mathbb{C})$ into $\mathrm{PSL}(n,\mathbb{C})$”, https://arxiv.org/abs/1412.3428, 2014
[11] Gerhard Burde, Heiner Zieschang & Michael Heusener, Knots, Berlin: Walter de Gruyter, 2013  MR 3156509 |  Zbl 1283.57002
[12] Marc Culler & Peter B. Shalen, “Varieties of group representations and splittings of $3$-manifolds”, Ann. Math. 117 (1983) no. 1, p. 109-146 Article |  MR 683804 |  Zbl 0529.57005
[13] Martin Deraux, “A 1-parameter family of spherical CR uniformizations of the figure eight knot complement”, https://arxiv.org/abs/1410.1198, 2014
[14] Martin Deraux, “On spherical CR uniformization of 3-manifolds”, https://arxiv.org/abs/1410.0659, 2014  MR 3359222
[15] Martin Deraux & Elisha Falbel, “Complex hyperbolic geometry of the figure-eight knot”, Geom. Topol. 19 (2015) no. 1, p. 237-293 Article |  MR 3318751
[16] Tudor T. Dimofte, M. Gabella & Alexander B. Goncharov, “K-Decompositions and 3d Gauge Theories”, https://arxiv.org/abs/1301.0192, 2013
[17] Tudor T. Dimofte & Stavros Garoufalidis, “The quantum content of the gluing equations”, https://arxiv.org/abs/1202.6268, 2012  MR 3073925
[18] Igor Dolgachev, Lectures on invariant theory, London Mathematical Society Lecture Note Series 296, Cambridge University Press, Cambridge, 2003 Article |  MR 2004511 |  Zbl 1023.13006
[19] Elisha Falbel, Antonin Guilloux, Pierre-Vincent Koseleff, Fabrice Rouillier & Morwen Thistlethwaite, “Character varieties for ${\rm SL}(3,\mathbb{C})$: the figure eight knot”, https://arxiv.org/abs/1412.4711, 2014  MR 3463570
[20] Vladimir Fock & Alexander B. Goncharov, “Moduli spaces of local systems and higher Teichmüller theory”, Publ. Math. Inst. Hautes Études Sci. (2006) no. 103, p. 1-211 Numdam |  MR 2233852 |  Zbl 1099.14025
[21] Charles D. Frohman & Eric Paul Klassen, “Deforming representations of knot groups in ${\rm SU}(2)$”, Comment. Math. Helv. 66 (1991) no. 3, p. 340-361 Article |  MR 1120651 |  Zbl 0738.57001
[22] William Fulton & Joe Harris, Representation theory, Graduate Texts in Mathematics 129, Springer-Verlag, 1991, A first course, Readings in Mathematics Article |  MR 1153249 |  Zbl 0744.22001
[23] Stavros Garoufalidis, Mattias Goerner & Christian K. Zickert, “Gluing equations for PGL(n,C)-representations of 3-manifolds”, https://arxiv.org/abs/1207.6711, 2012
[24] Stavros Garoufalidis, Dylan P. Thurston & Christian K. Zickert, “The complex volume of SL(n,C)-representations of 3-manifolds”, https://arxiv.org/abs/1111.2828, 2011  MR 3385130
[25] Stavros Garoufalidis & Christian K. Zickert, “The symplectic properties of the PGL(n,C)-gluing equations”, https://arxiv.org/abs/1310.2497, 2013
[26] William M. Goldman, “The symplectic nature of fundamental groups of surfaces”, Adv. Math. 54 (1984) no. 2, p. 200-225 Article |  MR 762512 |  Zbl 0574.32032
[27] Francisco González-Acuña & José María Montesinos-Amilibia, “On the character variety of group representations in ${\rm SL}(2,{\mathbb{C}})$ and ${\rm PSL}(2,{\mathbb{C}})$”, Math. Z. 214 (1993) no. 4, p. 627-652 Article |  Zbl 0799.20040
[28] Cameron Gordon, Dehn surgery and 3-manifolds, Low dimensional topology, IAS/Park City Math. Ser. 15, Amer. Math. Soc., Providence, RI, 2009, p. 21–71  MR 2503492 |  Zbl 1194.57003
[29] Christopher M. Herald, “Existence of irreducible representations for knot complements with nonconstant equivariant signature”, Math. Ann. 309 (1997) no. 1, p. 21-35 Article |  MR 1467643 |  Zbl 0887.57013
[30] Michael Heusener, “$\mathrm{SL}_n(\mathbb{C})$–representation spaces of knot groups”, RIMS Kôkyûroku (2016) no. 1991, p. 1-26, http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1991-01.pdf
[31] Michael Heusener & Jochen Kroll, “Deforming abelian ${\rm SU}(2)$-representations of knot groups”, Comment. Math. Helv. 73 (1998) no. 3, p. 480-498 Article |  MR 1633375 |  Zbl 0910.57004
[32] Michael Heusener & Ouardia Medjerab, “Deformations of reducible representations of knot groups into $\mathrm{SL}(n,\mathbf{C})$”, to appear in Mathematica Slovaca, https://arxiv.org/abs/1402.4294, 2014
[33] Michael Heusener, Vicente Muñoz & Joan Porti, “The ${\rm SL}(3,\mathbb{C})$-character variety of the figure eight knot.”, to appear in Illinois Journal of Mathematics, https://arxiv.org/abs/1505.04451, 2015
[34] Michael Heusener & Joan Porti, “Deformations of reducible representations of 3-manifold groups into ${\rm PSL}_2(\mathbb{C})$”, Algebr. Geom. Topol. 5 (2005), p. 965-997 Article |  MR 2171800 |  Zbl 1082.57007
[35] Michael Heusener & Joan Porti, “Representations of knot groups into $SL_n(\mathbb{C})$ and twisted Alexander polynomials”, Pacific J. Math. 277 (2015) no. 2, p. 313-354 Article |  MR 3402353
[36] Michael Heusener, Joan Porti & Eva Suárez Peiró, “Deformations of reducible representations of 3-manifold groups into ${\rm SL}_2(\mathbf{C})$”, J. Reine Angew. Math. 530 (2001), p. 191-227 Article |  MR 1807271 |  Zbl 0964.57006
[37] Irving Kaplansky, An introduction to differential algebra, Actualités Sci. Ind. 1251, Hermann, 1957  MR 93654 |  Zbl 0954.12500
[38] Michael Kapovich, Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009, Reprint of the 2001 edition Article |  MR 2553578 |  Zbl 1180.57001
[39] Michael Kapovich & John J. Millson, “On representation varieties of $3$-manifold groups”, https://arxiv.org/abs/1303.2347, 2013
[40] Paul Kirk & Charles Livingston, “Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants”, Topology 38 (1999) no. 3, p. 635-661 Article |  MR 1670420 |  Zbl 0928.57005
[41] Teruaki Kitano, “Twisted Alexander polynomial and Reidemeister torsion”, Pacific J. Math. 174 (1996) no. 2, p. 431-442  MR 1405595 |  Zbl 0863.57001
[42] Eric Paul Klassen, “Representations of knot groups in ${\rm SU}(2)$”, Trans. Amer. Math. Soc. 326 (1991) no. 2, p. 795-828 Article |  MR 1008696 |  Zbl 0743.57003
[43] Jerald J. Kovacic, “An algorithm for solving second order linear homogeneous differential equations”, J. Symbolic Comput. 2 (1986) no. 1, p. 3-43 Article |  MR 839134 |  Zbl 0603.68035
[44] Peter B. Kronheimer & Tomasz S. Mrowka, “Dehn surgery, the fundamental group and SU$(2)$”, Math. Res. Lett. 11 (2004) no. 5-6, p. 741-754 Article |  MR 2106239 |  Zbl 1084.57006
[45] Sean Lawton, “Generators, relations and symmetries in pairs of $3\times 3$ unimodular matrices”, J. Algebra 313 (2007) no. 2, p. 782-801 Article |  MR 2329569 |  Zbl 1119.13004
[46] Alexander Lubotzky & Andy R. Magid, “Varieties of representations of finitely generated groups”, Mem. Amer. Math. Soc. 58 (1985) no. 336 Article |  MR 818915 |  Zbl 0598.14042
[47] Pere Menal-Ferrer & Joan Porti, “Twisted cohomology for hyperbolic three manifolds”, Osaka J. Math. 49 (2012) no. 3, p. 741-769  MR 2993065 |  Zbl 1255.57018
[48] Pere Menal-Ferrer & Joan Porti, “Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds”, J. Topol. 7 (2014) no. 1, p. 69-119 Article |  MR 3180614 |  Zbl 1302.57044
[49] Werner Müller, The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, Progr. Math. 297, Birkhäuser/Springer, Basel, 2012, p. 317–352 Article |  MR 3220447 |  Zbl 1264.58026
[50] Vicente Muñoz & Joan Porti, “Geometry of the ${\rm SL}(3,\mathbb{C})$-character variety of torus knots”, https://arxiv.org/abs/1409.4784, 2014  MR 3470704
[51] Athanase Papadopoulos (éd.), Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics 13, European Mathematical Society (EMS), Zürich, 2009 Article
[52] Claudio Procesi, “The invariant theory of $n\times n$ matrices”, Adv. Math. 19 (1976) no. 3, p. 306-381  MR 419491 |  Zbl 0331.15021
[53] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, 1977, Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42  MR 450380 |  Zbl 0355.20006
[54] Douglas J. Shors, Deforming Reducible Representations of Knot Groups in $\rm {SL}(\mathbb{C})$., Ph. D. Thesis, U.C.L.A. (USA), 1991  MR 2686605
[55] Adam S. Sikora, “Character varieties”, Trans. Amer. Math. Soc. 364 (2012) no. 10, p. 5173-5208 Article |  MR 2931326 |  Zbl 1291.14022
[56] Tonny A. Springer, Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin, 1977  MR 447428 |  Zbl 0346.20020
[57] William P. Thurston, “The Geometry and Topology of Three-Manifolds”, Notes of Princeton University, http://library.msri.org/books/gt3m/, 1980
[58] William P. Thurston, “Three-dimensional manifolds, Kleinian groups and hyperbolic geometry”, Bull. Amer. Math. Soc. (N.S.) 6 (1982) no. 3, p. 357-381 Article |  MR 648524 |  Zbl 0496.57005
[59] Masaaki Wada, “Twisted Alexander polynomial for finitely presentable groups”, Topology 33 (1994) no. 2, p. 241-256 Article |  MR 1273784 |  Zbl 0822.57006
[60] Masaaki Wada, “Twisted Alexander polynomial revisited”, RIMS Kôkyûroku (2010) no. 1747, p. 140-144
[61] André Weil, “Remarks on the cohomology of groups”, Ann. Math. 80 (1964), p. 149-157  MR 169956 |  Zbl 0192.12802
[62] Pierre Will, Groupes libres, groupes triangulaires et tore épointé dans PU(2,1), Ph. D. Thesis, Université Pierre et Marie Curie – Paris VI (France), http://hal.archives-ouvertes.fr/tel-00130785v1, 2006
Copyright Cellule MathDoc 2019 | Crédit | Plan du site