$\mathrm{SL}_2$-character varieties of $2$-generated groups and failure of weak integrality

TL;DR

This paper investigates the $ ext{SL}_2$-character varieties of certain 2-generated groups, demonstrating failure of weak integrality at specific primes and providing a general description of related character varieties.

Contribution

It proves that the group $\Gamma_{ ext{ell}^k}$ is not $ ext{SL}_2$-weakly integral at prime $ ext{ell}$ and offers a broad characterization of character varieties for groups with specific relations.

Findings

$ ext{SL}_2$-weak integrality fails at prime $ ext{ell}$ for $\Gamma_{ ext{ell}^k}$.

Provides a general description of character varieties for 2-generated groups with certain relations.

Identifies the prime obstruction to weak integrality in these cases.

Abstract

Let $\ell$ be a prime number, $k$ a positive integer and consider the group $\Gamma_{\ell^k} :=\langle a,b\ \vert\ a^{\ell^k(\ell^k-1)}ba^{-\ell^k}b^{-2}\rangle$. We prove that $\Gamma_{\ell^k}$ is not $\mathrm{SL}_2$-weakly integral with obstruction at exactly the prime $\ell$. We also give a general description of the character varieties of $2$-generated groups with a relation of the form $a^{n_1} b^{m_1} a^{n_2} b^{m_2} = 1$.