Representation and character varieties of the Baumslag–Solitar groups

Representation and character varieties of the Baumslag–Solitar groups $\mathrm {BS}(p,q)$ are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety $R_n(\mathrm {BS}(p,q))$ are rational varieties of dimension $n^2$, and each irreducible component of the character variety $X_n(\mathrm {BS}(p,q))$ is a rational variety of dimension $k\le n$. The smoothness of irreducible components of the variety $R_n^\mathrm s(\mathrm {BS}(p,q))$ of irreducible representations is established, and it is proved that all irreducible components of the variety $X_n^\mathrm s(\mathrm {BS}(p,q))$ are isomorphic to $\mathbb {A}^1\setminus \{0\}$.