Revista de la
Unión Matemática Argentina
Coordinate rings of some $\mathrm{SL}_2$-character varieties
Vicente Muñoz and Jesús Martín Ovejero

Volume 67, no. 1 (2024), pp. 47–64    

Published online: March 8, 2024

https://doi.org/10.33044/revuma.3192

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Abstract

We determine generators of the coordinate ring of $\mathrm{SL}_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $\mathrm{SL}_2$-character variety. For free groups $F_k$, we find transcendental generators. Finally, for the case of the $2$-torus, we get an explicit equation of the $\mathrm{SL}_2$-character variety and use the description to compute their $E$-polynomials.

References

  1. C. Ashley, J.-P. Burelle, and S. Lawton, Rank 1 character varieties of finitely presented groups, Geom. Dedicata 192 (2018), 1–19.  DOI  MR  Zbl
  2. K. Corlette, Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 no. 3 (1988), 361–382.  MR  Zbl Available at http://projecteuclid.org/euclid.jdg/1214442469.
  3. M. Culler and P. B. Shalen, Varieties of group representations and splittings of $3$-manifolds, Ann. of Math. (2) 117 no. 1 (1983), 109–146.  DOI  MR  Zbl
  4. P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. no. 40 (1971), 5–57.  MR  Zbl Available at http://www.numdam.org/item?id=PMIHES_1971__40__5_0.
  5. C. Florentino and S. Lawton, Singularities of free group character varieties, Pacific J. Math. 260 no. 1 (2012), 149–179.  DOI  MR  Zbl
  6. C. Florentino, A. Nozad, and A. Zamora, Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups, J. Geom. Phys. 161 (2021), Paper No. 104008, 21 pp.  DOI  MR  Zbl
  7. W. M. Goldman, Trace coordinates on Fricke spaces of some simple hyperbolic surfaces, in Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys. 13, Eur. Math. Soc., Zürich, 2009, pp. 611–684.  DOI  MR  Zbl
  8. N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 no. 1 (1987), 59–126.  DOI  MR  Zbl
  9. T. Kitano and T. Morifuji, Twisted Alexander polynomials for irreducible $\mathrm{SL}(2,\mathbb{C})$-representations of torus knots, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 no. 2 (2012), 395–406.  DOI  MR  Zbl
  10. S. Lawton, Minimal affine coordinates for $\mathrm{SL}(3,\mathbb{C})$ character varieties of free groups, J. Algebra 320 no. 10 (2008), 3773–3810.  DOI  MR  Zbl
  11. S. Lawton and V. Muñoz, $E$-polynomial of the $\mathrm{SL}(3,\mathbb{C})$-character variety of free groups, Pacific J. Math. 282 no. 1 (2016), 173–202.  DOI  MR  Zbl
  12. J. D. Lewis, A survey of the Hodge conjecture, second ed., CRM Monograph Series 10, American Mathematical Society, Providence, RI, 1999.  DOI  MR  Zbl
  13. M. Logares, V. Muñoz, and P. E. Newstead, Hodge polynomials of $\mathrm{SL}(2,\mathbb{C})$-character varieties for curves of small genus, Rev. Mat. Complut. 26 no. 2 (2013), 635–703.  DOI  MR  Zbl
  14. A. Lubotzky and A. R. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 no. 336 (1985).  DOI  MR  Zbl
  15. J. Martín-Morales and A. M. Oller-Marcén, Combinatorial aspects of the character variety of a family of one-relator groups, Topology Appl. 156 no. 14 (2009), 2376–2389.  DOI  MR  Zbl
  16. J. Martínez and V. Muñoz, E-polynomials of $\mathrm{SL}(2,\mathbb{C})$-character varieties of complex curves of genus 3, Osaka J. Math. 53 no. 3 (2016), 645–681.  MR  Zbl Available at http://projecteuclid.org/euclid.ojm/1470413983.
  17. J. Martínez and V. Muñoz, E-polynomials of the $\mathrm{SL}(2,\mathbb{C})$-character varieties of surface groups, Int. Math. Res. Not. IMRN 2016 no. 3 (2016), 926–961.  DOI  MR  Zbl
  18. V. Muñoz, The $\mathrm{SL}(2,\mathbb{C})$-character varieties of torus knots, Rev. Mat. Complut. 22 no. 2 (2009), 489–497.  DOI  MR  Zbl
  19. V. Muñoz and J. Porti, Geometry of the $\mathrm{SL}(3,\mathbb{C})$-character variety of torus knots, Algebr. Geom. Topol. 16 no. 1 (2016), 397–426.  DOI  MR  Zbl
  20. C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. no. 79 (1994), 47–129.  DOI  MR  Zbl