Thin or arithmetic: symplectic hypergeometric groups
A hypergeometric group ⟨A, B⟩ ⊂ Sp_{2n}(Z) generated by two companion matrices is either of finite index (arithmetic) or thin. Decide the remaining cases, starting with degree 6.
Cite
@misc{cairn-thin-hypergeometric-groups,
title = {Thin or arithmetic: symplectic hypergeometric groups},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/thin-hypergeometric-groups}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
A Zariski-dense subgroup Γ of an arithmetic group G(Z) is thin if it has infinite index in G(Z), and arithmetic otherwise. For monic integer polynomials f, g of degree 2n with roots of unity as roots (and the usual conditions), the companion matrices A, B generate a hypergeometric group Γ(f, g) inside Sp_{2n}(Z) (or an orthogonal group). The session asked:
- Classify all pairs A, B ∈ GL_n(Z), n ≥ 3, for which ⟨A, B⟩ is thin.
- For hypergeometric groups: find a criterion for thinness comparable to the known criteria for arithmeticity (for example, arithmeticity follows when the leading nonzero coefficient of f − g is ±1 or ±2).
What is known
In degree 4 all symplectic hypergeometric groups have been decided. In degree 6 a 2026 preprint settles two further cases by exhibiting explicit witness words, and reports that one case (labelled C-32 in the Bajpai–Dona–Nitsche tables) remains undecided — a natural first target. General classification is wide open.
What counts as progress
- Arithmeticity certificates: explicit words in A, B satisfying a published arithmeticity criterion, verified in exact integer arithmetic.
- Thinness proofs: ping-pong configurations or other arguments, with all computations reproducible.
- New criteria for thinness, tested against the decided cases.
How it is checked
Arithmeticity certificates are exact computations (level B: reproducible scripts in Sage/Magma/GAP). Thinness arguments are reviewed.
Source. Posed by Jitendra Bajpai in the open problem session of the Oberwolfach workshop Mini-Workshop: Growth and Expansion in Groups (2024), recorded in Oberwolfach Reports 17/2024, p. 1035 (EMS Press, DOI 10.4171/OWR/2024/17), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.