Failure of the congruence Hasse principle for commutators in orientable surface groups

Abstract

Ghosh, Meiri and Sarnak showed that the commutator equation over $\mathrm{SL}_2(\mathbb{Z})$ satisfies a profinite local to global principle, and that for a prime $\ell\equiv\pm1$ mod $5$ it fails for $\mathrm{SL}_2(\mathbb{Z}[1/\ell])$ with a reciprocity obstruction. We extend the failure to surface groups. For the genus-$2$ surface group $Γ_2$ we construct an explicit representation $ρ:Γ_2\to\mathrm{SL}_2(\mathcal{O}_{K,S})$, where $K=\mathbb{Q}(\sqrt{5})$ and $\mathcal{O}_{K,S}=\mathbb{Z}[\frac{1+\sqrt{5}}{2},\frac{1}{10}]$, whose quotients modulo the maximal ideals are the full groups $\mathrm{SL}_2(\mathbb{F}_q)$ except in residue characteristic $3$, where the quotient has order $24$. For every $k\ge2$ the word $[a_1,b_1]^k$ is not a commutator in $Γ_2$, while its image is a commutator in every one of these quotients. The same holds for every surface group $Γ_g$ with $g\ge2$.