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Eulerian cube complexes and reciprocity

Richard Scott

Algebraic & Geometric Topology 14 (2014) 3533–3552

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) = gt(g), where (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(t1) = (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.

cube complex, growth series
Mathematical Subject Classification 2000
Primary: 20F55
Secondary: 20F10, 05A15
Received: 4 October 2013
Revised: 19 February 2014
Accepted: 23 February 2014
Published: 15 January 2015
Richard Scott
Mathematics and Computer Science
Santa Clara University 500 El Camino Real
Santa Clara, CA 95053