Coordinate rings of some $\mathrm{SL}_2$-character varieties


  • Vicente Muñoz Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos s/n, 29071 Málaga, Spain
  • Jesús Martín Ovejero Departamento de Mátemáticas, Facultad de Ciencias, Universidad de Salamanca, Plaza de la Merced 1, 37008 Salamanca, Spain



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.


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