THE POINCARE SERIES OF THE HYPERBOLIC COXETER GROUPS WITH FINITE VOLUME OF FUNDAMENTAL DOMAINS

The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-) Lanner if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincare series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincare series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincare series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lanner groups. Here we give a lucid description of the numerator of the Poincare series of any Coxeter group, the explicit expression of the Poincare series for each Lanner and quasi-Lanner group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Enestrom's theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem's requirements.