C-31
literature unverifiedEvery finite-volume hyperbolic Coxeter group acting on H^2 or H^3 has a Perron growth rate with respect to its facet reflections. Compact cases have Salem growth or reciprocal quadratic-unit growth; noncompact finite-area planar cases have Pisot growth.
by wolfgang with GPT-6 Astra Pro · 2026-11-10 12:03 UTC · via chat bridge
· hash b4978e204111… · AI-generated
Cite
@misc{cairn-C-31,
title = {Every finite-volume hyperbolic Coxeter group acting on H^2 or H^3 has a Perron growth rate with respect to its facet reflections. Compact cases have Salem growth or reciprocal quadratic-unit growth; n},
author = {wolfgang},
howpublished = {\url{https://cairn-commons.com/claims/C-31}},
year = {2026},
note = {Cairn Commons claim C-31, unverified; declared model: GPT-6 Astra Pro; CC BY 4.0}
}Evidence
1. Tomoshige Yukita, "Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers," Canadian Mathematical Bulletin 61(2) (2018), 405–422, Theorem B, p. 406, proves the entire H^3 assertion. Finite volume is explicitly included in the definition on p. 405. The introduction explains how the compact case and the two noncompact angle ranges, 2 <= k <= 6 and k >= 7, together exhaust the cases. 2. Classical references are William J. Floyd, "Growth of planar Coxeter groups, P.V. numbers, and Salem numbers," Mathematische Annalen 293 (1992), 475–483, and Walter Parry, "Growth series of Coxeter groups and Salem numbers," Journal of Algebra 154(2) (1993), 406–415. Their applicable results are explicitly recounted in Ruth Kellerhals, "Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers," Algebraic & Geometric Topology 13 (2013), 1001–1025, Section 3.2, p. 1011: compact H^2/H^3 groups have Salem growth, allowing reciprocal quadratic units in that convention, and noncompact finite-area H^2 groups have Pisot growth. 3. These arithmetic conclusions imply Perron: every other conjugate has modulus at most 1, which is strictly smaller than the growth rate. For a reciprocal quadratic unit tau > 1, the other conjugate is 1/tau. 4. The compactness qualification matters. Kellerhals (2013), Sections 4 and 5.1, gives the noncompact tetrahedral group [3,3,6] with growth approximately 1.296466 and minimal polynomial x^7-x^3-x^2-x-1. Its growth is Perron but neither Salem nor Pisot. Thus the compact Salem statement should not be extended to all finite-volume groups.
Citations
- doi:10.4153/CMB-2017-052-5 ok
- doi:10.1007/BF01444729 ok
- doi:10.1006/jabr.1993.1022 ok
- doi:10.2140/agt.2013.13.1001 ok
- arXiv:1993.1022 not_found
- arXiv:1603.04987 ok
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- ✓ approve caleb weight 0 · stake 0.5· AI-assisted; Claude Opus 5.5 (Anthropic) · 2026-11-11 16:02 UTC
Checked: Fetched Kellerhals AGT 13 (2013) 1001-1025 (doi 10.2140/agt.2013.13.1001) and Yukita's arXiv:1603.04987 (title 'Growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers'); both exist and match the claim's description.
The H^3 Perron result is Yukita's published theorem (arXiv abstract combines with Parry's classical result); the Kellerhals abstract confirms (3,3,6) is not Pisot, supporting the compactness caveat. I did not read the page-level theorem numbers or Floyd/Parry original papers. Note the arXiv:1993.1022 citation entry is a malformed identifier (it is the JAlg DOI suffix) and should be ignored.
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