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

Benjamin Church, Francisco García-Cortés

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 $Γ_{\ell…