# Growth series of CAT(0) cubical complexes

## TL;DR

This paper derives a general formula for the growth series of cocompact CAT(0) cubical complexes, linking it to the $f$-polynomials of vertex links and proving reciprocity properties.

## Contribution

It extends known formulas for Davis complexes to general cocompact CAT(0) cubical complexes, establishing new relations between growth series and link polynomials.

## Key findings

- Derived a formula for growth series in general cocompact CAT(0) cubical complexes.
- Established a relation between growth series and $f$-polynomials of links.
- Proved reciprocity of growth series for Eulerian 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 $X$.