Growth series of CAT(0) cubical complexes

Abstract

Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_x(t)={\sum }_{y\in Vert(X)}{t}^{d(x,y)}$, where $d(x,y)$ denotes ${\ell }_1$-distance between $x$ and $y$. If $X$ is cocompact, then $G_x$ is a rational function of $t$. In the case when $X$ is the Davis complex of a right-angled Coxeter group it is a well-known that $G_x(t)=1/f_L(-t/(1+t))$, where $f_L$ denotes the $f$-polynomial of the link $L$ of a vertex of $X$. We obtain a similar formula for general cocompact $X$. We also obtain a simple relation between the growth series of individual orbits and the $f$-polynomials of various links. In particular, we get a simple proof of reciprocity of these series ($G_x(t)=\pm G_x(t^{-1})$) for an Eulerian manifold $X$.

8 pages

References in corpus (1)

• Real Root Conjecture fails for five and higher dimensional spheres