Are growth rates of hyperbolic Coxeter groups Perron numbers?
Conjecture (Kellerhals–Perren): the growth rate of every cocompact hyperbolic Coxeter group is a Perron number. Testable polytope by polytope with CoxIter.
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@misc{cairn-coxeter-growth-rates-perron,
title = {Are growth rates of hyperbolic Coxeter groups Perron numbers?},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/coxeter-growth-rates-perron}},
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
The growth rate of a Coxeter group with respect to its standard generators is the inverse of the radius of convergence of its growth series. Conjecture (Kellerhals–Perren): the growth rate of every cocompact hyperbolic Coxeter group is a Perron number — a real algebraic integer > 1 strictly larger in absolute value than all its other Galois conjugates. The abstract notes the conjecture seems to hold also for finite covolume.
What is known
In dimensions 2 and 3 the growth rates are Salem numbers (a special case); the triangle group (2, 3, 7) gives Lehmer's number. In dimension ≥ 4 cocompact growth rates are not Salem numbers. Not every Salem number occurs as a growth rate of a cocompact hyperbolic Coxeter group.
What counts as progress
- Systematic computation (e.g. with CoxIter) of growth rates for all known compact and finite-volume hyperbolic Coxeter polytopes in dimensions 4–8, with a check of the Perron property of each minimal polynomial.
- A counterexample: one Coxeter polytope whose growth rate has a Galois conjugate of equal or larger modulus.
- Proofs for infinite families.
How it is checked
Growth series are rational functions computable exactly; the Perron property is an exact check on the minimal polynomial. Scripts and output tables must be reproducible (level B).
Source. Posed by Ruth Kellerhals in an extended abstract of the Oberwolfach workshop Explicit Methods in Number Theory (2024), recorded in Oberwolfach Reports 40/2024, p. 2328 (EMS Press, DOI 10.4171/OWR/2024/40), 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.