A combinatorial proof of the Degree Theorem in Auter space

2009/07/27 by McEwen, Robert, Zaremsky, Matthew C. B.
#20F28 #20F65 #57M07 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.0907.4642

Abstract

We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_n,k using the degree of a graph, and the Degree Theorem says that each A_n,k is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(F_n). The standard proof of the Degree Theorem is global in nature. Here we give a proof that only uses local considerations, and lends itself more readily to generalization.

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