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The discrete group generated by reflections of the sphere, or Euclideanspace, or hyperbolic space are said to be Coxeter groups of, respectively,spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups aresaid to be (quasi-)Lann\'er if the tiles covering the space are of finitevolume and all (resp. some of them) are compact. For any Coxeter groupstratified by the length of its elements, the Poincar\'e series (a.k.a. growthfunction) is the generating function of the cardinalities of sets of elementsof equal length. Solomon established that, for ANY Coxeter group, itsPoincar\'e series is a rational function with zeros somewhere on the unitcircle centered at the origin, and gave a recurrence formula. The explicitexpression of the Poincar\'e series was known for the spherical and EuclideanCoxeter groups, and 3-generated Coxeter groups, and (with mistakes) Lann\'ergroups. Here we give a lucid description of the numerator of the Poincar\'eseries of any Coxeter group, and denominators for each (quasi-)Lann\'er group,and review the scene. We give an interpretation of some coefficients of thedenominator of the Poincar\'e series. The non-real poles behave as inEnestr\"om's theorem (lie in a narrow annulus) though the coefficients of thedenominators do not satisfy theorem's requirements.

The Poincare series of the hyperbolic Coxeter groups with finite volume of fundamental domains