On representation varieties of some HNN-extensions of free groups

In the article we provide the description of the structure and the properties of representation varieties $R_{n}(G(p,q))$ of the groups with the presentation $G(p,q)=\langle x_{1},\dots , x_{2},t|t(x_{1}^{2}\dots x_{g}^{2})=(x_{1}^{2}\dots x_{g}^{2})^{q}\rangle$, where $g\geq 3, |p|>q\geq 1$. Irreducible components of $R_{n}(G(p,q))$ are found, their dimensions are calculated and it is proved, that every irreducible component of $R_{n}(G(p,q))$ is a rational variety.