Some recent results about the SL n ()–representation spaces of knot groups
Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 137-161.

This survey reviews some facts about about the representation and character varieties of knot groups into SL n () with n3 are presented. This concerns mostly joint work of the author with L. Ben Abdelghani, O. Medjerab, V. Muños and J. Porti.

DOI : 10.5802/tsg.307
Classification : 57M25, 57M05, 57M27
Mots clés : knot group, representation variety, character variety
Heusener, Michael 1, 2

1 Université Clermont Auvergne Université Blaise Pascal Laboratoire de Mathématiques BP 10448 63000 Clermont-Ferrand (France)
2 and CNRS, UMR 6620, LM 63178 Aubiere (France)
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Heusener, Michael. Some recent results about the $\mathrm{SL}_n(\mathbb{C})$–representation spaces of knot groups. Séminaire de théorie spectrale et géométrie, Tome 32 (2014-2015), pp. 137-161. doi : 10.5802/tsg.307. http://archive.numdam.org/articles/10.5802/tsg.307/

[1] Artin, Michael On the solutions of analytic equations, Invent. Math., Volume 5 (1968), pp. 277-291 | MR | Zbl

[2] Ben Abdelghani, Leila Espace des représentations du groupe d’un nœud classique dans un groupe de Lie, Ann. Inst. Fourier (Grenoble), Volume 50 (2000) no. 4, pp. 1297-1321 | Numdam | MR | Zbl

[3] Ben Abdelghani, Leila Tangent cones and local geometry of the representation and character varieties of knot groups, Algebr. Geom. Topol., Volume 10 (2010) no. 1, pp. 433-463 | DOI | MR | Zbl

[4] Ben Abdelghani, Leila; Heusener, Michael Irreducible representations of knot groups into SL(n,C) (2015) (to appear in Publicacions Matemàtiques, https://arxiv.org/abs/1111.2828)

[5] Ben Abdelghani, Leila; Heusener, Michael; Jebali, Hajer Deformations of metabelian representations of knot groups into SL (3,C), J. Knot Theory Ramifications, Volume 19 (2010) no. 3, pp. 385-404 | DOI | MR | Zbl

[6] Bergeron, Nicolas; Falbel, Elisha; Guilloux, Antonin Tetrahedra of flags, volume and homology of SL (3), Geom. Topol., Volume 18 (2014) no. 4, pp. 1911-1971 | DOI | MR

[7] Boden, Hans U.; Friedl, Stefan Metabelian SL (n,) representations of knot groups, Pacific J. Math., Volume 238 (2008) no. 1, pp. 7-25 | DOI | MR | Zbl

[8] Boden, Hans U.; Friedl, Stefan Metabelian SL (n,) representations of knot groups, III: deformations, Q. J. Math., Volume 65 (2014) no. 3, pp. 817-840 | MR

[9] Brown, Kenneth S. Cohomology of groups, Graduate Texts in Mathematics, 87, Springer-Verlag, 1994, pp. x+306 (Corrected reprint of the 1982 original) | MR | Zbl

[10] Bucher, Michelle; Burger, Marc; Iozzi, Alessandra Rigidity of representations of hyperbolic lattices Γ<PSL(2,) into PSL(n,) (2014) (https://arxiv.org/abs/1412.3428)

[11] Burde, Gerhard; Zieschang, Heiner; Heusener, Michael Knots, Berlin: Walter de Gruyter, 2013, pp. xiii + 417 | MR | Zbl

[12] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of 3-manifolds, Ann. Math., Volume 117 (1983) no. 1, pp. 109-146 | DOI | MR | Zbl

[13] Deraux, Martin A 1-parameter family of spherical CR uniformizations of the figure eight knot complement (2014) (https://arxiv.org/abs/1410.1198)

[14] Deraux, Martin On spherical CR uniformization of 3-manifolds (2014) (https://arxiv.org/abs/1410.0659) | MR

[15] Deraux, Martin; Falbel, Elisha Complex hyperbolic geometry of the figure-eight knot, Geom. Topol., Volume 19 (2015) no. 1, pp. 237-293 | DOI | MR

[16] Dimofte, Tudor T.; Gabella, M.; Goncharov, Alexander B. K-Decompositions and 3d Gauge Theories (2013) (https://arxiv.org/abs/1301.0192)

[17] Dimofte, Tudor T.; Garoufalidis, Stavros The quantum content of the gluing equations (2012) (https://arxiv.org/abs/1202.6268) | MR

[18] Dolgachev, Igor Lectures on invariant theory, London Mathematical Society Lecture Note Series, 296, Cambridge University Press, Cambridge, 2003, pp. xvi+220 | DOI | MR | Zbl

[19] Falbel, Elisha; Guilloux, Antonin; Koseleff, Pierre-Vincent; Rouillier, Fabrice; Thistlethwaite, Morwen Character varieties for SL (3,): the figure eight knot (2014) (https://arxiv.org/abs/1412.4711) | MR

[20] Fock, Vladimir; Goncharov, Alexander B. Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. (2006) no. 103, pp. 1-211 | DOI | Numdam | MR | Zbl

[21] Frohman, Charles D.; Klassen, Eric Paul Deforming representations of knot groups in SU (2), Comment. Math. Helv., Volume 66 (1991) no. 3, pp. 340-361 | DOI | MR | Zbl

[22] Fulton, William; Harris, Joe Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, 1991, pp. xvi+551 (A first course, Readings in Mathematics) | DOI | MR | Zbl

[23] Garoufalidis, Stavros; Goerner, Mattias; Zickert, Christian K. Gluing equations for PGL(n,C)-representations of 3-manifolds (2012) (https://arxiv.org/abs/1207.6711)

[24] Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, Christian K. The complex volume of SL(n,C)-representations of 3-manifolds (2011) (https://arxiv.org/abs/1111.2828) | MR

[25] Garoufalidis, Stavros; Zickert, Christian K. The symplectic properties of the PGL(n,C)-gluing equations (2013) (https://arxiv.org/abs/1310.2497)

[26] Goldman, William M. The symplectic nature of fundamental groups of surfaces, Adv. Math., Volume 54 (1984) no. 2, pp. 200-225 | DOI | MR | Zbl

[27] González-Acuña, Francisco; Montesinos-Amilibia, José María On the character variety of group representations in SL (2,) and PSL (2,), Math. Z., Volume 214 (1993) no. 4, pp. 627-652 | DOI | Zbl

[28] Gordon, Cameron Dehn surgery and 3-manifolds, Low dimensional topology (IAS/Park City Math. Ser.), Volume 15, Amer. Math. Soc., Providence, RI, 2009, pp. 21-71 | MR | Zbl

[29] Herald, Christopher M. Existence of irreducible representations for knot complements with nonconstant equivariant signature, Math. Ann., Volume 309 (1997) no. 1, pp. 21-35 | DOI | MR | Zbl

[30] Heusener, Michael SL n ()–representation spaces of knot groups, RIMS Kôkyûroku (2016) no. 1991, pp. 1-26 (http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1991-01.pdf)

[31] Heusener, Michael; Kroll, Jochen Deforming abelian SU (2)-representations of knot groups, Comment. Math. Helv., Volume 73 (1998) no. 3, pp. 480-498 | DOI | MR | Zbl

[32] Heusener, Michael; Medjerab, Ouardia Deformations of reducible representations of knot groups into SL(n,C) (2014) (to appear in Mathematica Slovaca, https://arxiv.org/abs/1402.4294)

[33] Heusener, Michael; Muñoz, Vicente; Porti, Joan The SL (3,)-character variety of the figure eight knot. (2015) (to appear in Illinois Journal of Mathematics, https://arxiv.org/abs/1505.04451)

[34] Heusener, Michael; Porti, Joan Deformations of reducible representations of 3-manifold groups into PSL 2 (), Algebr. Geom. Topol., Volume 5 (2005), pp. 965-997 | DOI | MR | Zbl

[35] Heusener, Michael; Porti, Joan Representations of knot groups into SL n () and twisted Alexander polynomials, Pacific J. Math., Volume 277 (2015) no. 2, pp. 313-354 | DOI | MR

[36] Heusener, Michael; Porti, Joan; Suárez Peiró, Eva Deformations of reducible representations of 3-manifold groups into SL 2 (C), J. Reine Angew. Math., Volume 530 (2001), pp. 191-227 | DOI | MR | Zbl

[37] Kaplansky, Irving An introduction to differential algebra, Actualités Sci. Ind., 1251, Hermann, 1957, pp. 63 | MR | Zbl

[38] Kapovich, Michael Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009, pp. xxviii+467 (Reprint of the 2001 edition) | DOI | MR | Zbl

[39] Kapovich, Michael; Millson, John J. On representation varieties of 3-manifold groups (2013) (https://arxiv.org/abs/1303.2347)

[40] Kirk, Paul; Livingston, Charles Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology, Volume 38 (1999) no. 3, pp. 635-661 | DOI | MR | Zbl

[41] Kitano, Teruaki Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., Volume 174 (1996) no. 2, pp. 431-442 http://projecteuclid.org/getRecord?id=euclid.pjm/1102365178 | MR | Zbl

[42] Klassen, Eric Paul Representations of knot groups in SU (2), Trans. Amer. Math. Soc., Volume 326 (1991) no. 2, pp. 795-828 | DOI | MR | Zbl

[43] Kovacic, Jerald J. An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput., Volume 2 (1986) no. 1, pp. 3-43 | DOI | MR | Zbl

[44] Kronheimer, Peter B.; Mrowka, Tomasz S. Dehn surgery, the fundamental group and SU(2), Math. Res. Lett., Volume 11 (2004) no. 5-6, pp. 741-754 | DOI | MR | Zbl

[45] Lawton, Sean Generators, relations and symmetries in pairs of 3×3 unimodular matrices, J. Algebra, Volume 313 (2007) no. 2, pp. 782-801 | DOI | MR | Zbl

[46] Lubotzky, Alexander; Magid, Andy R. Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc., Volume 58 (1985) no. 336, pp. xi+117 | DOI | MR | Zbl

[47] Menal-Ferrer, Pere; Porti, Joan Twisted cohomology for hyperbolic three manifolds, Osaka J. Math., Volume 49 (2012) no. 3, pp. 741-769 | MR | Zbl

[48] Menal-Ferrer, Pere; Porti, Joan Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol., Volume 7 (2014) no. 1, pp. 69-119 | DOI | MR | Zbl

[49] Müller, Werner The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry (Progr. Math.), Volume 297, Birkhäuser/Springer, Basel, 2012, pp. 317-352 | DOI | MR | Zbl

[50] Muñoz, Vicente; Porti, Joan Geometry of the SL (3,)-character variety of torus knots (2014) (https://arxiv.org/abs/1409.4784) | MR

[51] Papadopoulos, Athanase Handbook of Teichmüller theory. Vol. II, IRMA Lectures in Mathematics and Theoretical Physics, 13, European Mathematical Society (EMS), Zürich, 2009, pp. x+874 | DOI

[52] Procesi, Claudio The invariant theory of n×n matrices, Adv. Math., Volume 19 (1976) no. 3, pp. 306-381 | MR | Zbl

[53] Serre, Jean-Pierre Linear representations of finite groups, Springer-Verlag, 1977, pp. x+170 (Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, Vol. 42) | MR | Zbl

[54] Shors, Douglas J. Deforming Reducible Representations of Knot Groups in SL ()., U.C.L.A. (USA) (1991) (Ph. D. Thesis) | MR

[55] Sikora, Adam S. Character varieties, Trans. Amer. Math. Soc., Volume 364 (2012) no. 10, pp. 5173-5208 | DOI | MR | Zbl

[56] Springer, Tonny A. Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin, 1977, pp. iv+112 | MR | Zbl

[57] Thurston, William P. The Geometry and Topology of Three-Manifolds (1980) (Notes of Princeton University, http://library.msri.org/books/gt3m/)

[58] Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), Volume 6 (1982) no. 3, pp. 357-381 | DOI | MR | Zbl

[59] Wada, Masaaki Twisted Alexander polynomial for finitely presentable groups, Topology, Volume 33 (1994) no. 2, pp. 241-256 | DOI | MR | Zbl

[60] Wada, Masaaki Twisted Alexander polynomial revisited, RIMS Kôkyûroku (2010) no. 1747, pp. 140-144

[61] Weil, André Remarks on the cohomology of groups, Ann. Math., Volume 80 (1964), pp. 149-157 | MR | Zbl

[62] Will, Pierre Groupes libres, groupes triangulaires et tore épointé dans PU(2,1), Université Pierre et Marie Curie – Paris VI (France) (2006) (Ph. D. Thesis http://hal.archives-ouvertes.fr/tel-00130785v1)

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