Some recent results about the ${\mathrm{SL}}_{n}\left(ℂ\right)$–representation spaces of knot groups
Séminaire de théorie spectrale et géométrie, Volume 32  (2014-2015), p. 137-161

This survey reviews some facts about about the representation and character varieties of knot groups into ${\mathrm{SL}}_{n}\left(ℂ\right)$ 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.

DOI : https://doi.org/10.5802/tsg.307
Classification:  57M25,  57M05,  57M27
Keywords: knot group, representation variety, character variety
@article{TSG_2014-2015__32__137_0,
author = {Heusener, Michael},
title = {Some recent results about the $\mathrm{SL}\_n(\mathbb{C})$--representation spaces of knot groups},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
publisher = {Institut Fourier},
volume = {32},
year = {2014-2015},
pages = {137-161},
doi = {10.5802/tsg.307},
language = {en},
url = {http://www.numdam.org/item/TSG_2014-2015__32__137_0}
}

Heusener, Michael. Some recent results about the $\mathrm{SL}_n(\mathbb{C})$–representation spaces of knot groups. Séminaire de théorie spectrale et géométrie, Volume 32 (2014-2015) , pp. 137-161. doi : 10.5802/tsg.307. http://www.numdam.org/item/TSG_2014-2015__32__137_0/

[1] Artin, Michael On the solutions of analytic equations, Invent. Math., Tome 5 (1968), pp. 277-291 | MR 232018 | Zbl 0172.05301

[2] Ben Abdelghani, Leila Espace des représentations du groupe d’un nœud classique dans un groupe de Lie, Ann. Inst. Fourier (Grenoble), Tome 50 (2000) no. 4, pp. 1297-1321 | MR 1799747 | Zbl 0956.57006

[3] Ben Abdelghani, Leila Tangent cones and local geometry of the representation and character varieties of knot groups, Algebr. Geom. Topol., Tome 10 (2010) no. 1, pp. 433-463 | Article | MR 2602842 | Zbl 1201.57002

[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 $\mathrm{SL}\left(3,\mathbf{C}\right)$, J. Knot Theory Ramifications, Tome 19 (2010) no. 3, pp. 385-404 | Article | MR 2646638 | Zbl 1195.57023

[6] Bergeron, Nicolas; Falbel, Elisha; Guilloux, Antonin Tetrahedra of flags, volume and homology of $\mathrm{SL}\left(3\right)$, Geom. Topol., Tome 18 (2014) no. 4, pp. 1911-1971 | Article | MR 3268771

[7] Boden, Hans U.; Friedl, Stefan Metabelian $\mathrm{SL}\left(n,ℂ\right)$ representations of knot groups, Pacific J. Math., Tome 238 (2008) no. 1, pp. 7-25 | Article | MR 2443505 | Zbl 1154.57004

[8] Boden, Hans U.; Friedl, Stefan Metabelian $\mathrm{SL}\left(n,ℂ\right)$ representations of knot groups, III: deformations, Q. J. Math., Tome 65 (2014) no. 3, pp. 817-840 | MR 3261968

[9] Brown, Kenneth S. Cohomology of groups, Springer-Verlag, Graduate Texts in Mathematics, Tome 87 (1994), pp. x+306 (Corrected reprint of the 1982 original) | MR 1324339 | Zbl 0584.20036

[10] Bucher, Michelle; Burger, Marc; Iozzi, Alessandra Rigidity of representations of hyperbolic lattices $\Gamma <\mathrm{PSL}\left(2,ℂ\right)$ into $\mathrm{PSL}\left(n,ℂ\right)$ (2014) (https://arxiv.org/abs/1412.3428)

[11] Burde, Gerhard; Zieschang, Heiner; Heusener, Michael Knots, Berlin: Walter de Gruyter (2013), pp. xiii + 417 | MR 3156509 | Zbl 1283.57002

[12] Culler, Marc; Shalen, Peter B. Varieties of group representations and splittings of $3$-manifolds, Ann. Math., Tome 117 (1983) no. 1, pp. 109-146 | Article | MR 683804 | Zbl 0529.57005

[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 3359222

[15] Deraux, Martin; Falbel, Elisha Complex hyperbolic geometry of the figure-eight knot, Geom. Topol., Tome 19 (2015) no. 1, pp. 237-293 | Article | MR 3318751

[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 3073925

[18] Dolgachev, Igor Lectures on invariant theory, Cambridge University Press, Cambridge, London Mathematical Society Lecture Note Series, Tome 296 (2003), pp. xvi+220 | Article | MR 2004511 | Zbl 1023.13006

[19] Falbel, Elisha; Guilloux, Antonin; Koseleff, Pierre-Vincent; Rouillier, Fabrice; Thistlethwaite, Morwen Character varieties for $\mathrm{SL}\left(3,ℂ\right)$: the figure eight knot (2014) (https://arxiv.org/abs/1412.4711) | MR 3463570

[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 | Article | Numdam | MR 2233852 | Zbl 1099.14025

[21] Frohman, Charles D.; Klassen, Eric Paul Deforming representations of knot groups in $\mathrm{SU}\left(2\right)$, Comment. Math. Helv., Tome 66 (1991) no. 3, pp. 340-361 | Article | MR 1120651 | Zbl 0738.57001

[22] Fulton, William; Harris, Joe Representation theory, Springer-Verlag, Graduate Texts in Mathematics, Tome 129 (1991), pp. xvi+551 (A first course, Readings in Mathematics) | Article | MR 1153249 | Zbl 0744.22001

[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 3385130

[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., Tome 54 (1984) no. 2, pp. 200-225 | Article | MR 762512 | Zbl 0574.32032

[27] González-Acuña, Francisco; Montesinos-Amilibia, José María On the character variety of group representations in $\mathrm{SL}\left(2,ℂ\right)$ and $\mathrm{PSL}\left(2,ℂ\right)$, Math. Z., Tome 214 (1993) no. 4, pp. 627-652 | Article | Zbl 0799.20040

[28] Gordon, Cameron Dehn surgery and 3-manifolds, Low dimensional topology, Amer. Math. Soc., Providence, RI (IAS/Park City Math. Ser.) Tome 15 (2009), pp. 21-71 | MR 2503492 | Zbl 1194.57003

[29] Herald, Christopher M. Existence of irreducible representations for knot complements with nonconstant equivariant signature, Math. Ann., Tome 309 (1997) no. 1, pp. 21-35 | Article | MR 1467643 | Zbl 0887.57013

[30] Heusener, Michael ${\mathrm{SL}}_{n}\left(ℂ\right)$–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 $\mathrm{SU}\left(2\right)$-representations of knot groups, Comment. Math. Helv., Tome 73 (1998) no. 3, pp. 480-498 | Article | MR 1633375 | Zbl 0910.57004

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

[33] Heusener, Michael; Muñoz, Vicente; Porti, Joan The $\mathrm{SL}\left(3,ℂ\right)$-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 ${\mathrm{PSL}}_{2}\left(ℂ\right)$, Algebr. Geom. Topol., Tome 5 (2005), pp. 965-997 | Article | MR 2171800 | Zbl 1082.57007

[35] Heusener, Michael; Porti, Joan Representations of knot groups into $S{L}_{n}\left(ℂ\right)$ and twisted Alexander polynomials, Pacific J. Math., Tome 277 (2015) no. 2, pp. 313-354 | Article | MR 3402353

[36] Heusener, Michael; Porti, Joan; Suárez Peiró, Eva Deformations of reducible representations of 3-manifold groups into ${\mathrm{SL}}_{2}\left(\mathbf{C}\right)$, J. Reine Angew. Math., Tome 530 (2001), pp. 191-227 | Article | MR 1807271 | Zbl 0964.57006

[37] Kaplansky, Irving An introduction to differential algebra, Hermann, Actualités Sci. Ind., Tome 1251 (1957), pp. 63 | MR 93654 | Zbl 0954.12500

[38] Kapovich, Michael Hyperbolic manifolds and discrete groups, Birkhäuser Boston, Inc., Boston, MA, Modern Birkhäuser Classics (2009), pp. xxviii+467 (Reprint of the 2001 edition) | Article | MR 2553578 | Zbl 1180.57001

[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, Tome 38 (1999) no. 3, pp. 635-661 | Article | MR 1670420 | Zbl 0928.57005

[41] Kitano, Teruaki Twisted Alexander polynomial and Reidemeister torsion, Pacific J. Math., Tome 174 (1996) no. 2, pp. 431-442 http://projecteuclid.org/getRecord?id=euclid.pjm/1102365178 | MR 1405595 | Zbl 0863.57001

[42] Klassen, Eric Paul Representations of knot groups in $\mathrm{SU}\left(2\right)$, Trans. Amer. Math. Soc., Tome 326 (1991) no. 2, pp. 795-828 | Article | MR 1008696 | Zbl 0743.57003

[43] Kovacic, Jerald J. An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Comput., Tome 2 (1986) no. 1, pp. 3-43 | Article | MR 839134 | Zbl 0603.68035

[44] Kronheimer, Peter B.; Mrowka, Tomasz S. Dehn surgery, the fundamental group and SU$\left(2\right)$, Math. Res. Lett., Tome 11 (2004) no. 5-6, pp. 741-754 | Article | MR 2106239 | Zbl 1084.57006

[45] Lawton, Sean Generators, relations and symmetries in pairs of $3×3$ unimodular matrices, J. Algebra, Tome 313 (2007) no. 2, pp. 782-801 | Article | MR 2329569 | Zbl 1119.13004

[46] Lubotzky, Alexander; Magid, Andy R. Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc., Tome 58 (1985) no. 336, pp. xi+117 | Article | MR 818915 | Zbl 0598.14042

[47] Menal-Ferrer, Pere; Porti, Joan Twisted cohomology for hyperbolic three manifolds, Osaka J. Math., Tome 49 (2012) no. 3, pp. 741-769 | MR 2993065 | Zbl 1255.57018

[48] Menal-Ferrer, Pere; Porti, Joan Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds, J. Topol., Tome 7 (2014) no. 1, pp. 69-119 | Article | MR 3180614 | Zbl 1302.57044

[49] Müller, Werner The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, Birkhäuser/Springer, Basel (Progr. Math.) Tome 297 (2012), pp. 317-352 | Article | MR 3220447 | Zbl 1264.58026

[50] Muñoz, Vicente; Porti, Joan Geometry of the $\mathrm{SL}\left(3,ℂ\right)$-character variety of torus knots (2014) (https://arxiv.org/abs/1409.4784) | MR 3470704

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

[52] Procesi, Claudio The invariant theory of $n×n$ matrices, Adv. Math., Tome 19 (1976) no. 3, pp. 306-381 | MR 419491 | Zbl 0331.15021

[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 450380 | Zbl 0355.20006

[54] Shors, Douglas J. Deforming Reducible Representations of Knot Groups in $\mathrm{SL}\left(ℂ\right)$., U.C.L.A. (USA) (1991) (Ph. D. Thesis) | MR 2686605

[55] Sikora, Adam S. Character varieties, Trans. Amer. Math. Soc., Tome 364 (2012) no. 10, pp. 5173-5208 | Article | MR 2931326 | Zbl 1291.14022

[56] Springer, Tonny A. Invariant theory, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 585 (1977), pp. iv+112 | MR 447428 | Zbl 0346.20020

[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.), Tome 6 (1982) no. 3, pp. 357-381 | Article | MR 648524 | Zbl 0496.57005

[59] Wada, Masaaki Twisted Alexander polynomial for finitely presentable groups, Topology, Tome 33 (1994) no. 2, pp. 241-256 | Article | MR 1273784 | Zbl 0822.57006

[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., Tome 80 (1964), pp. 149-157 | MR 169956 | Zbl 0192.12802

[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)