Reidemeister torsion form on character varieties

Bénard, Léo¹

¹Mathematisches Institut, Georg-August Unversität, Göttingen, Germany

Algebraic and Geometric Topology, Tome 20 (2020) no. 6, pp. 2821-2884

Résumé

We define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3–manifold with toral boundary, and prove it defines a rational volume form. Then we show that the torsion form has poles only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion has no pole under a generic hypothesis on the Alexander polynomial; otherwise, we relate the order of the pole with the type of singularity. Finally we consider the ideal points added after compactification of the character variety. We bound the vanishing order of the torsion by the Euler characteristic of an essential surface associated to the ideal point by the Culler–Shalen theory. As a corollary we obtain an unexpected relation between the topology of those surfaces and the topology of the character variety.

Détail

Classification : 57M25, 57M27
Keywords: Reidemeister torsion, character varieties, $3$–manifolds, Culler–Shalen theory

Bibliographie

[1] L Bénard, Torsion function on character varieties, preprint (2017)

[2] L Bénard, Reidemeister torsion on character varieties, PhD thesis, Sorbonne Université (2018)

[3] L Bénard, A Conway, A multivariable Casson–Lin type invariant, Ann. Inst. Fourier (Grenoble) 70 (2020) 1029 | DOI

[4] R W Berger, Report on the torsion of the differential module of an algebraic curve, from: "Algebraic geometry and its applications" (editor C L Bajaj), Springer (1994) 285 | DOI

[5] K S Brown, Cohomology of groups, 87, Springer (1982) | DOI

[6] G Burde, Darstellungen von Knotengruppen, Math. Ann. 173 (1967) 24 | DOI

[7] T A Chapman, Topological invariance of Whitehead torsion, Amer. J. Math. 96 (1974) 488 | DOI

[8] M Chu, Detecting essential surfaces as intersections in the character variety, Algebr. Geom. Topol. 17 (2017) 2893 | DOI

[9] M M Cohen, A course in simple-homotopy theory, 10, Springer (1973) | DOI

[10] D Cooper, M Culler, H Gillet, D D Long, P B Shalen, Plane curves associated to character varieties of 3–manifolds, Invent. Math. 118 (1994) 47 | DOI

[11] M Culler, C M Gordon, J Luecke, P B Shalen, Dehn surgery on knots, Ann. of Math. 125 (1987) 237 | DOI

[12] M Culler, P B Shalen, Varieties of group representations and splittings of 3–manifolds, Ann. of Math. 117 (1983) 109 | DOI

[13] M Culler, P B Shalen, Bounded, separating, incompressible surfaces in knot manifolds, Invent. Math. 75 (1984) 537 | DOI

[14] J F Davis, P Kirk, Lecture notes in algebraic topology, 35, Amer. Math. Soc. (2001) | DOI

[15] J Dubois, Torsion de Reidemeister non abélienne et forme volume sur l’espace des représentations du groupe d’un nœud, PhD thesis, Université Blaise Pascal, Clermont-Ferrand II (2003)

[16] J Dubois, Non abelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull. 49 (2006) 55 | DOI

[17] J Dubois, S Garoufalidis, Rationality of the SL(2, C)–Reidemeister torsion in dimension 3, Topology Proc. 47 (2016) 115

[18] J Dubois, V Huynh, Y Yamaguchi, Non-abelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications 18 (2009) 303 | DOI

[19] N M Dunfield, S Friedl, N Jackson, Twisted Alexander polynomials of hyperbolic knots, Exp. Math. 21 (2012) 329 | DOI

[20] S Friedl, T Kitayama, M Nagel, Representation varieties detect essential surfaces, Math. Res. Lett. 25 (2018) 803 | DOI

[21] C D Frohman, J Kania-Bartoszynska, Dubois torsion, A–polynomial and quantum invariants, Quantum Topol. 4 (2013) 187 | DOI

[22] W Fulton, Algebraic curves: an introduction to algebraic geometry, Benjamin (1969)

[23] I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser (1994) | DOI

[24] A Hatcher, W Thurston, Incompressible surfaces in 2–bridge knot complements, Invent. Math. 79 (1985) 225 | DOI

[25] M Heusener, J Porti, Representations of knot groups into SLn(C) and twisted Alexander polynomials, Pacific J. Math. 277 (2015) 313 | DOI

[26] M Heusener, J Porti, E Suárez Peiró, Deformations of reducible representations of 3–manifold groups into SL2(C), J. Reine Angew. Math. 530 (2001) 191 | DOI

[27] D Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann. 36 (1890) 473 | DOI

[28] T Kitano, Reidemeister torsion of Seifert fibered spaces for SL(2; C)–representations, Tokyo J. Math. 17 (1994) 59 | DOI

[29] T Kitano, Reidemeister torsion of the figure-eight knot exterior for SL(2; C)–representations, Osaka J. Math. 31 (1994) 523

[30] H Kraft, C Procesi, Classical invariant theory: a primer, preprint (1996)

[31] Q Liu, Algebraic geometry and arithmetic curves, 6, Oxford Univ. Press (2002)

[32] A Lubotzky, A R Magid, Varieties of representations of finitely generated groups, 336, Amer. Math. Soc. (1985) | DOI

[33] J Marché, Character varieties in SL2 and Kauffman skein algebras, preprint (2015)

[34] J Marché, M Wolff, The parallelogram identity on groups and deformations of the trivial character in SL2(C), J. Éc. Polytech. Math. 7 (2020) 263 | DOI

[35] J Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358 | DOI

[36] D Mumford, J Fogarty, F Kirwan, Geometric invariant theory, 34, Springer (1994)

[37] W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307 | DOI

[38] J Park, Half-density volumes of representation spaces of some 3–manifolds and their application, Duke Math. J. 86 (1997) 493 | DOI

[39] J Porti, Torsion de Reidemeister pour les variétés hyperboliques, 612, Amer. Math. Soc. (1997) | DOI

[40] C Procesi, A formal inverse to the Cayley–Hamilton theorem, J. Algebra 107 (1987) 63 | DOI

[41] J H Przytycki, Incompressibility of surfaces after Dehn surgery, Michigan Math. J. 30 (1983) 289 | DOI

[42] J H Przytycki, A S Sikora, On skein algebras and Sl2(C)–character varieties, Topology 39 (2000) 115 | DOI

[43] G De Rham, Introduction aux polynômes d’un nœud, Enseign. Math. 13 (1967) 187

[44] K Saito, Character variety of representations of a finitely generated group in SL2, from: "Topology and Teichmüller spaces" (editors S Kojima, Y Matsumoto, K Saito, M Seppälä), World Sci. (1996) 253 | DOI

[45] J P Serre, Arbres, amalgames, SL2, 46, Soc. Math. France (1977)

[46] P B Shalen, Representations of 3–manifold groups, from: "Handbook of geometric topology" (editors R J Daverman, R B Sher), North-Holland (2002) 955

[47] W P Thurston, Three-dimensional geometry and topology, I, 35, Princeton Univ. Press (1997)

[48] S Tillmann, Finding hyperbolic structures of and surfaces in 3–manifolds, course notes (2003)

[49] A T Tran, Twisted Alexander polynomials of genus one two-bridge knots, Kodai Math. J. 41 (2018) 86 | DOI

[50] V Turaev, Introduction to combinatorial torsions, Birkhäuser (2001) | DOI

[51] A Weil, Remarks on the cohomology of groups, Ann. of Math. 80 (1964) 149 | DOI

[52] E Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991) 153 | DOI

[53] S Yoon, A vanishing identity on adjoint Reidemeister torsions, preprint (2020)