Coordinate rings of some $SL_2$-character varieties

Abstract

We determine generators of the coordinate ring of $SL_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $SL_2$-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 $SL_2$-character variety and use the description to compute their E-polynomials.