Geodesic
Eulerian cube complexes and reciprocity
Scott, Richard1
1Mathematics and Computer Science, Santa Clara University 500 El Camino Real, Santa Clara, CA 95053, USA
Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3533-3552
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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 Gx(t) = ∑ gtℓ(g), where ℓ(g) denotes the length of g. Using the fact that G admits a suitable automatic structure, Gx(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 Gx(t−1) = (−1)nGx(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.

DOI : 10.2140/agt.2014.14.3533
Keywords: cube complex, growth series

Scott, Richard  1

1 Mathematics and Computer Science, Santa Clara University 500 El Camino Real, Santa Clara, CA 95053, USA 
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Scott, Richard. Eulerian cube complexes and reciprocity. Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3533-3552. doi: 10.2140/agt.2014.14.3533

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