#### Volume 14, issue 6 (2014)

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Eulerian cube complexes and reciprocity

### Richard Scott

Algebraic & Geometric Topology 14 (2014) 3533–3552
##### Abstract

Let $G$ be the fundamental group of a compact nonpositively curved cube complex $Y$. With respect to a basepoint $x$, one obtains an integer-valued length function on $G$ by counting the number of edges in a minimal length edge-path representing each group element. The growth series of $G$ with respect to $x$ is then defined to be the power series ${G}_{x}\left(t\right)={\sum }_{g}{t}^{\ell \left(g\right)}$, where $\ell \left(g\right)$ denotes the length of $g$. Using the fact that $G$ admits a suitable automatic structure, ${G}_{x}\left(t\right)$ can be shown to be a rational function. We prove that if $Y$ is a manifold of dimension $n$, then this rational function satisfies the reciprocity formula ${G}_{x}\left({t}^{-1}\right)={\left(-1\right)}^{n}{G}_{x}\left(t\right)$. We prove the formula in a more general setting, replacing the group with the fundamental groupoid, replacing the growth series with the characteristic series for a suitable regular language, and only assuming $Y$ is Eulerian.

##### Keywords
cube complex, growth series
##### Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 20F10, 05A15