Growth of planar Coxeter groups, P.V.numbers and Salem numbers (Q811511)

Let \(D\) be a finite sided polygon in the hyperbolic plane \(\mathbb{H}^ 2\) such that each vertex of \(D\) is either an ideal vertex or a vertex in \(\mathbb{H}^ 2\) with angle measure a submultiple of \(\pi\). Let \(\Sigma\) be the set of reflections in the edges of \(D\), and let \(G\) be the subgroup of \(\hbox{Isom}(\mathbb{H}^ 2)\) generated by \(\Sigma\). Then \(G\) is a Coxeter group and \((G,\Sigma)\) is a Coxeter system. Let \(\alpha=\alpha_{G,D}\) be the growth exponent of \(G\) with respect to the generating set \(\Sigma\). We prove that \(\alpha\) is a P. V. number if \(D\) has an ideal vertex. If \(D\) has \(k\geq 2\) ideal vertices, then \(\alpha\) is in the \((k- 1)^{st}\) derived set of the set of P. V. numbers. If \(D\) has an ideal vertex and \(D_ r\) is the polygon obtained by ''replacing'' each ideal vertex of \(D\) by a vertex with angle measure \(\pi/r\), then the resulting growth exponents \(\alpha_ r\) are the standard sequence of Salem numbers converging to \(\alpha\) from below.