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

Abstract

Let $\ell$ be a prime number, $k$ a positive integer and consider the group ${\Gamma }_{{\ell }^k}:=⟨a,b|a^{{\ell }^k({\ell }^k-1)}b{a}^{-{\ell }^k}{b}^{-2}⟩$. 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 $a^{n_1}{b}^{m_1}{a}^{n_2}{b}^{m_2}=1$.

Correct two typos on the abstract, improve one of main results, add references