Data about hyperbolic Coxeter systems

Abstract

We collect several data about Coxeter systems (cf. [Bou07, Hum90]), with particular emphasis on the hyperbolic ones. For each ($\preceq$-minimal) hyperbolic Coxeter system (W,S) the Poincar\'e series \[p_{(W,S)}(t)=\sum_{w\in W} t^{\ell(w)}\] and the growth rate \[ \omega(W,S)=\limsup_n \sqrt[n]{a_n}\] are explicitly computed using Magma (cf. [BCP97]). These computations were performed in connection to the proof of [Ter, Thm. B]. Since the Poincar\'e series represents a rational function, one may recover the sequence $(a_k)_{k\geq 0}$ through a linear recurrence relation on the coefficients, provided that enough terms at the beginning of the sequence are known. For each Coxeter system the initial coefficients $(a_k)_{k=0}^N$ are computed, where $N$ is the degree of the numerator of $p_{(W,S)}(t)$.