On the varieties of representations and characters of a family of one-relator subgroups. Their irreducible components

TL;DR

This paper explicitly describes the structure of the character variety of certain one-relator groups, providing formulas for irreducible components and methods to recover group parameters from geometric data.

Contribution

It offers an explicit primary decomposition of the character variety for groups defined by a single relator, and relates geometric structure to group parameters.

Findings

Formula for the number of irreducible components of the character variety.

Explicit primary decomposition of the character variety.

Method to recover group parameters from the combinatorial structure.

Abstract

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible to recover $(m,n)$ from the combinatorial structure of $X(G)$. Finally we compute the number of irreducible components of $R(G)$ and study the behavior of the projection $t:R(G)\longrightarrow X(G)$.