Abstract
Let be the fundamental group of a compact nonpositively curved cube complex . With respect to a basepoint , one obtains an integer-valued length function on by counting the number of edges in a minimal length edge-path representing each group element. The growth series of with respect to is then defined to be the power series , where denotes the length of . Using the fact that admits a suitable automatic structure, can be shown to be a rational function. We prove that if is a manifold of dimension , then this rational function satisfies the reciprocity formula . 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 is Eulerian.
Citation
Richard Scott. "Eulerian cube complexes and reciprocity." Algebr. Geom. Topol. 14 (6) 3533 - 3552, 2014. https://doi.org/10.2140/agt.2014.14.3533
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