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Failure of the congruence Hasse principle for commutators in orientable surface groups
Abstract
Ghosh, Meiri and Sarnak showed that the commutator equation over SL2(ℤ) satisfies a profinite local to global principle, and that for a prime ℓ≡±1 mod $5$ it fails for SL2(ℤ[1/ℓ]) with a reciprocity obstruction. We extend the failure to surface groups. For the genus-$2$ surface group Γ2 we construct an explicit representation ρ:Γ2→SL2(𝒪K,S), where K=ℚ(√5) and 𝒪K,S=ℤ[(1+√5)/2,1/10], whose quotients modulo the maximal ideals are the full groups SL2(𝔽q) except in residue characteristic $3$, where the quotient has order $24$. For every k≥2 the word [a1,b1]k is not a commutator in Γ2, while its image is a commutator in every one of these quotients. The same holds for every surface group Γg with g≥2.
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BibTeX
@misc{wu2026failure,
title = {{Failure of the congruence Hasse principle for commutators in orientable surface groups}},
author = {Jianchun Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36497},
eprint = {2610.36497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36497},
}
RIS
TY - UNPB
TI - Failure of the congruence Hasse principle for commutators in orientable surface groups
AU - Jianchun Wu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36497
UR - https://arxiv.org/abs/2610.36497
AB - Ghosh, Meiri and Sarnak showed that the commutator equation over SL_2(ℤ) satisfies a profinite local to global principle, and that for a prime ℓ≡±1 mod $5$ it fails for SL_2(ℤ[1/ℓ]) 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→SL_2(𝒪_K,S), where K=ℚ(√5) and 𝒪_K,S=ℤ[(1+√5)/2,1/10], whose quotients modulo the maximal ideals are the full groups SL_2(𝔽_q) except in residue characteristic $3$, where the quotient has order $24$. For every k≥2 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≥2.
ER -