Multivariate growth series of graph products of groups
Right-angled Artin groups (RAAGs) and right-angled Coxeter groups (RACGs) associated with finite simple graphs are fundamental objects in geometric group theory. Their one-variable growth series with respect to the standard generating sets was classically expressed by Chiswell in terms of the one-variable independence polynomial of the defining graph [2] with suitable substitutions of the variable. In this paper, we investigate the multivariate growth series of graph products of groups and derive explicit formulas in terms of the multivariate independence polynomial of the underlying graph through suitable substitutions of variables. As special cases, we obtain multivariate growth series formulas for RAAGs and RACGs, thereby extending the classical one-variable identities. We further show that the coefficients of the multivariate growth series of RAAGs and RACGs admit explicit descriptions in terms of the double-marked and marked chromatic polynomials of graphs. This connection reveals a rich interplay between growth series and graph coloring invariants. In particular, we obtain completely explicit formulas for all the coefficients in the case of chordal graphs, which include, for example, trees and complete graphs.