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OP-10213Status unassessedNumber theory

Decidability

Arithmetic reflection groups and crystallographic packings · AIMPL workshop record

MSC 11 · Number theory

Editorial mapping of AIMPL source category: Number theory. http://aimpl.org/reflectiongpv/ Not an author-supplied MSC code.

Workshop record · status unassessedAs of 2026-09-10 source snapshot; not a current-status certificationSource & evidence ↓
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Problem statement

Is reflectivity of a maximal arithmetic lattice a decidable problem? If so, can we find an efficient algorithm to solve this decision problem?

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Background

Section context

In this section, Γ<O⁡(n,1)\Gamma < \operatorname{O}(n,1) is taken to be an arithmetic lattice. We'll write Γ=R⋊Aut⁡(P)\Gamma = R \rtimes \operatorname{Aut}(P) where RR is generated by reflections and Aut⁡(P)\operatorname{Aut}(P) is the group of symmetries of the Coxeter polytope PP corresponding to RR.

Question context

Vinberg's algorithm terminates if Γ\Gamma is reflective (i.e. if ∣Aut⁡(P)∣<∞|\operatorname{Aut}(P)| < \infty), and does not terminate if Γ\Gamma is not reflective. As such, determining whether such a given arithmetic lattice is reflective is at least semi-decidable.

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Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

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References
  1. [1]Arithmetic reflection groups and crystallographic packings · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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