On the growth of a Coxeter group

For a Coxeter system (W,S) let a(n)((W,S)) be the cardinality of the sphere of radius n in the Cayley graph of W with respect to the standard generating set S. It is shown that, if (W,S)<=(W',S') then a(n)((W,S))=a(n)((W',S')) for all n is an element of N-0, where <= is a suitable partial order on Coxeter systems (cf.Thm.A). It is proven that there exists a constant tau=1.13...such that for any non-affine, non-spherical Coxeter system (W,S) the growth rate omega(W,S)=lim sup(n)root a(n) satisfies omega(W,S)>= t (cf.Thm.B). The constant t is a Perron number of degree 127 over Q. For a Coxeter group W the Coxeter generating set is not unique (up to W-conjugacy), but there is a standard procedure, the diagram twisting (cf. [3]), which allows one to pass from one Coxeter generating set S to another Coxeter generating set mu(S). A generalisation of the diagram twisting is introduced, the mutation, and it is proven that Poincare series are invariant under mutations (cf. Thm. C).