Those are notes of the mini-course given in the school Winter Braids VII, held in Caen from February 27th to March 2nd 2017. They overview the variety of representations and characters of a three-manifold in , putting emphasis on explicit computations. The notes also discuss the canonical component of a hyperbolic knot, and a recent joint work with Luisa Paoluzzi, on the invariant components of the variety of characters for knot symmetries.
@article{WBLN_2017__4__A2_0, author = {Joan Porti}, title = {Character varieties and knot symmetries}, journal = {Winter Braids Lecture Notes}, note = {talk:2}, pages = {1--21}, publisher = {Winter Braids School}, volume = {4}, year = {2017}, doi = {10.5802/wbln.18}, zbl = {07113760}, mrnumber = {3922034}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.18/} }
TY - JOUR AU - Joan Porti TI - Character varieties and knot symmetries JO - Winter Braids Lecture Notes N1 - talk:2 PY - 2017 SP - 1 EP - 21 VL - 4 PB - Winter Braids School UR - https://proceedings.centre-mersenne.org/articles/10.5802/wbln.18/ DO - 10.5802/wbln.18 LA - en ID - WBLN_2017__4__A2_0 ER -
Joan Porti. Character varieties and knot symmetries. Winter Braids Lecture Notes, Volume 4 (2017), Talk no. 2, 21 p. doi : 10.5802/wbln.18. https://proceedings.centre-mersenne.org/articles/10.5802/wbln.18/
[1] Gérard Besson. Calabi-Weil infinitesimal rigidity. In Géométries à courbure négative ou nulle, groupes discrets et rigidités, volume 18 of Sémin. Congr., pages 177–200. Soc. Math. France, Paris, 2009.
[2] S. Boyer and X. Zhang. On Culler-Shalen seminorms and Dehn filling. Ann. of Math. (2), 148(3):737–801, 1998. | DOI | Zbl
[3] Steven Boyer and Xingru Zhang. Every nontrivial knot in has nontrivial -polynomial. Proc. Amer. Math. Soc., 133(9):2813–2815, 2005. | DOI | Zbl
[4] K. Bromberg. Rigidity of geometrically finite hyperbolic cone-manifolds. Geom. Dedicata, 105:143–170, 2004. | DOI | Zbl
[5] Gerhard Burde. -representation spaces for two-bridge knot groups. Math. Ann., 288(1):103–119, 1990. | DOI | Zbl
[6] Richard D. Canary. Dynamics on character varieties: a survey. In Handbook of group actions. Vol. II, volume 32 of Adv. Lect. Math. (ALM), pages 175–200. Int. Press, Somerville, MA, 2015.
[7] Alex Casella, Feng Luo, and Stephan Tillmann. Pseudo-developing maps for ideal triangulations II: Positively oriented ideal triangulations of cone-manifolds. Proc. Amer. Math. Soc., 145(8):3543–3560, 2017. | Zbl
[8] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen. Plane curves associated to character varieties of -manifolds. Invent. Math., 118(1):47–84, 1994. | DOI | Zbl
[9] Marc Culler. Lifting representations to covering groups. Adv. in Math., 59(1):64–70, 1986. | DOI | Zbl
[10] Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen. Dehn surgery on knots. Ann. of Math. (2), 125(2):237–300, 1987. | DOI | Zbl
[11] Marc Culler and Peter B. Shalen. Varieties of group representations and splittings of -manifolds. Ann. of Math. (2), 117(1):109–146, 1983. | Zbl
[12] Igor Dolgachev. Lectures on invariant theory, volume 296 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003. | Zbl
[13] Nathan M. Dunfield and Stavros Garoufalidis. Non-triviality of the -polynomial for knots in . Algebr. Geom. Topol., 4:1145–1153, 2004. | DOI | Zbl
[14] W. Goldman. Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pages 611–684. Eur. Math. Soc., Zürich, 2009. | Zbl
[15] F. González-Acuña and José María Montesinos-Amilibia. On the character variety of group representations in and . Math. Z., 214(4):627–652, 1993. | DOI | Zbl
[16] Shinya Harada. Canonical components of character varieties of arithmetic two bridge link complements. arXiv:1112.3441, 2011. | Zbl
[17] Hugh M. Hilden, María Teresa Lozano, and José Marí a Montesinos-Amilibia. On the character variety of periodic knots and links. Math. Proc. Cambridge Philos. Soc., 129(3):477–490, 2000. | DOI | Zbl
[18] Craig D. Hodgson. Degeneration and Regeneration of Geometric Structures on Three-Manifolds, 1986. PhD Thesis, Princeton University.
[19] Michael Kapovich. Hyperbolic manifolds and discrete groups, volume 183 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2001. | Zbl
[20] P. B. Kronheimer and T. S. Mrowka. Dehn surgery, the fundamental group and SU. Math. Res. Lett., 11(5-6):741–754, 2004. | Zbl
[21] Alexander Lubotzky and Andy R. Magid. Varieties of representations of finitely generated groups. Mem. Amer. Math. Soc., 58(336):xi+117, 1985. | Zbl
[22] Melissa L. Macasieb, Kathleen L. Petersen, and Ronald M. van Luijk. On character varieties of two-bridge knot groups. Proc. Lond. Math. Soc. (3), 103(3):473–507, 2011. | Zbl
[23] Colin Maclachlan and Alan W. Reid. The arithmetic of hyperbolic 3-manifolds, volume 219 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2003. | DOI
[24] Wilhelm Magnus. Rings of Fricke characters and automorphism groups of free groups. Math. Z., 170(1):91–103, 1980. | DOI | Zbl
[25] D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, 1994. | Zbl
[26] Walter D. Neumann and Don Zagier. Volumes of hyperbolic three-manifolds. Topology, 24(3):307–332, 1985. | Zbl
[27] P. E. Newstead. Introduction to moduli problems and orbit spaces, volume 51 of Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978.
[28] Tomotada Ohtsuki, Robert Riley, and Makoto Sakuma. Epimorphisms between 2-bridge link groups. In The Zieschang Gedenkschrift, volume 14 of Geom. Topol. Monogr., pages 417–450. Geom. Topol. Publ., Coventry, 2008. | Zbl
[29] Luisa Paoluzzi and Joan Porti. Non-standard components of the character variety for a family of Montesinos knots. Proc. Lond. Math. Soc. (3), 107(3):655–679, 2013. | DOI | Zbl
[30] Luisa Paoluzzi and Joan Porti. Invariant character varieties of hyperbolic knots with symmetries. Mathematical Proceedings of the Cambridge Philosophical Society 165 (2018), no. 2, 193–208. | Zbl
[31] K. L. Petersen and A. W. Reid. Gonality and genus of canonical components of character varieties. ArXiv e-prints, August 2014.
[32] Joan Porti. Torsion de Reidemeister pour les variétés hyperboliques. Mem. Amer. Math. Soc., 128(612):x+139, 1997. | DOI | Zbl
[33] C. Procesi. The invariant theory of matrices. Advances in Math., 19(3):306–381, 1976. | DOI | Zbl
[34] Robert Riley. Parabolic representations of knot groups. I. Proc. London Math. Soc. (3), 24:217–242, 1972. | DOI | Zbl
[35] Robert Riley. Parabolic representations of knot groups. II. Proc. London Math. Soc. (3), 31(4):495–512, 1975. | DOI | Zbl
[36] Robert Riley. Nonabelian representations of -bridge knot groups. Quart. J. Math. Oxford Ser. (2), 35(138):191–208, 1984. | DOI | Zbl
[37] Adam S. Sikora. Character varieties. Trans. Amer. Math. Soc., 364(10):5173–5208, 2012. | DOI | Zbl
[38] T. A. Springer. Invariant theory. Lecture Notes in Mathematics, Vol. 585. Springer-Verlag, Berlin, 1977. | DOI | Zbl
[39] William P. Thurston. The Geometry and Topology of Three-Manifolds. Princeton University, http://www.msri.org/publications/books/gt3m/. | Zbl
[40] È. B. Vinberg and V. L. Popov. Invariant theory. In Algebraic geometry, 4 (Russian), Itogi Nauki i Tekhniki, pages 137–314, 315. Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989.
[41] Henry Vogt. Sur les invariants fondamentaux des équations différentielles linéaires du second ordre. Ann. Sci. Éc. Norm. Supér., 6:3–71, 1889. | DOI | Zbl
[42] André Weil. Remarks on the cohomology of groups. Ann. of Math. (2), 80:149–157, 1964. | DOI | Zbl
[43] Alice Whittemore. On representations of the group of Listing’s knot by subgroups of . Proc. Amer. Math. Soc., 40:378–382, 1973. | DOI | Zbl
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