On representation varieties of some one-relator products of cyclic groups

Abstract

In the paper representation varieties of two classes of finitely generated groups are investigated.
The first class consists of groups with the presentation
\begin{gather*}
G = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g\mid\\
a_1^{m_1}=\ldots=a_s^{m_s}= x_1^2\ldots x_g^2 W(a_1,\ldots,a_s,b_1,\ldots,b_k)=1\rangle,
\end{gather*}
where $g\ge 3$, $m_i\ge 2$ for $i=1,\ldots,s$ and $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ is an element in normal form in the free product of cyclic groups
$$
H=\langle a_1\mid a_1^{m_1}\rangle\ast\ldots\ast\langle a_s\mid a_s^{m_s}\rangle\ast\langle b_1\rangle\ast\ldots\ast
\langle b_k\rangle.
$$

The second class consists of groups with the presentation
$$
G(p,q) = \langle a_1,\ldots,a_s,b_1,\ldots,b_k,x_1,\ldots,x_g,t\mid a_1^{m_1}=\ldots=a_s^{m_s}=1,\ tU^pt^{-1}=U^q \rangle,
$$
where $p$ and $q$ are integer numbers such that $p>|q|\geq1$, $(p,q)=1$, $m_i\ge 2$ for $i=1,\ldots,s$, \linebreak $g\ge3$,
$U=x_1^2\ldots x_g^2W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ and $W(a_1,\ldots,a_s,b_1,\ldots,b_k)$ is an above defined element.

Irreducible components of representation varieties $R_n(G)$ and $R_n(G(p,q))$ are found, their dimensions are calculated and it is proved, that every irreducible component is a rational variety.