Coordinate rings of some $SL_2$-character varieties

TL;DR

This paper explicitly describes the generators and equations of $SL_2$-character varieties for certain groups, providing new algebraic descriptions and computations of their invariants.

Contribution

It determines generators and explicit equations for $SL_2$-character varieties of free groups and the 2-torus, including transcendental generators and E-polynomial computations.

Findings

Explicit generators for $SL_2$-character varieties of free groups and the 2-torus.

Derived explicit equations for these varieties.

Computed E-polynomials for the 2-torus case.

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.