Coxeter systems with \(2\)-dimensional Davis complexes, growth rates and Perron numbers (Q6593015)

Let \((W,S)\) be a Coxeter system. Denote by \(a_{\ell}\) the number of elements in \(W\) whose word lengths are \(\ell\). The growth rate of \((W,S)\) is defined to be\N\[\N\limsup_{\ell\rightarrow \infty}\sqrt[\ell]{a_{\ell}}.\N\]\NThe growth rate is the inverse of the radius of convergence of the growth series \(f_{(W,S)}(z)=\sum_{\ell\geq 0}a_{\ell}z^{\ell}.\) It is known that the growth rate of an affine Coxeter system is 1. For many classes of hyperbolic Coxeter systems, it was previously shown by various authors that the growth rates are either Salem numbers, Pisot numbers, or other Perron numbers (those that are not necessarily Salem numbers or Pisot numbers).\N\NIn the paper under review, the authors investigate the growth rate of a non-spherical, non-affine Coxeter system whose Davis complex has dimension at most \(2\) by relating the growth rate to the Euler characteristic of the nerve of \((W,S)\). The authors show that if the Euler characteristic is \(0\) (resp. positive), then the growth rate is a Salem (resp. Pisot) number. Furthermore, when the Euler characteristic is positive, the authors show that there exists a sequence of Coxeter systems \((W_n, S_n)\) (with dimensions at most \(2\) and vanishing Euler characteristics) such that the growth rates of \((W_n, S_n)\) converge to that of \((W, S)\) from below. Finally, the authors construct infinitely many non-hyperbolic Coxeter systems with dimension at most \(2\) and negative Euler characteristics, whose growth rates are Perron numbers.