Eulerian cube complexes and reciprocity

Scott, Richard

doi:10.2140/agt.2014.14.3533

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Eulerian cube complexes and reciprocity

Richard Scott

Algebraic & Geometric Topology 14 (2014) 3533–3552

DOI: 10.2140/agt.2014.14.3533

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 $G_{x} (t) = \sum_{g} t^{ℓ (g)}$ , where $ℓ (g)$ denotes the length of $g$ . Using the fact that $G$ admits a suitable automatic structure, $G_{x} (t)$ 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 $G_{x} (t^{- 1}) = {(- 1)}^{n} G_{x} (t)$ . 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

References

Publication

Received: 4 October 2013

Revised: 19 February 2014

Accepted: 23 February 2014

Published: 15 January 2015

Authors

Richard Scott
Mathematics and Computer Science Santa Clara University 500 El Camino Real Santa Clara, CA 95053 USA

Volume 14, issue 6 (2014)