Salem numbers, spectral radii and growth rates of hyperbolic Coxeter groups

We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups.

💡 Research Summary

The paper investigates the interplay between Salem numbers, spectral radii of Coxeter transformations, and growth rates of hyperbolic Coxeter groups. After recalling the basic definitions of a compact hyperbolic Coxeter polyhedron (P\subset\mathbb H^{n}), its reflection group (G), and the growth series (f_{S}(t)=\sum_{w\in W}t^{\ell_{S}(w)}), the authors note that for compact hyperbolic groups the growth series is a rational function whose radius of convergence (R) yields the growth rate (\tau=1/R). Classical results of Floyd, Plotnick and Parry imply that for dimensions (n=2,3) the growth rate is either a Salem number or a quadratic unit.

The first main result (Theorem 1) shows that not every Salem number can arise as the growth rate of a compact hyperbolic Coxeter polyhedron. The proof proceeds by examining the natural geometric representation (\rho:W\to GL(V)) of an abstract Coxeter system ((W,S)) and the associated Coxeter element (c) of length (|S|). Its image (C=\rho(c)) is the Coxeter transformation. Using the eigenvalue structure of (C) and the relationship between the leading eigenvalue (\alpha) of the Coxeter adjacency matrix and the spectral radius (\lambda) of (C) (namely (\alpha^{2}=2+\lambda+\lambda^{-1})), the authors derive necessary algebraic conditions for (\lambda) to be a Salem number. They then exhibit families of star graphs (\operatorname{Star}(p_{1},\dots ,p_{k})) whose Coxeter transformations fail to satisfy these conditions for infinitely many Salem numbers, thereby proving the non‑realizability claim.

The second main result (Theorem 2) establishes a precise equality between growth rates and spectral radii in the planar case. For a compact hyperbolic Coxeter (k)-gon ((p_{1},\dots ,p_{k})) with (\sum 1/p_{i}<k-2), the growth function can be written explicitly as

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