New York Journal of Mathematics
Volume 20 (2014) 217-228


Robert McEwen and Matthew C. B. Zaremsky

A combinatorial proof of the Degree Theorem in Auter space

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Published: March 4, 2014
Keywords: Auter space, Degree Theorem, automorphisms of free groups
Subject: Primary 20F65; Secondary 57M07, 20F28

We use discrete Morse theory to give a new proof of Hatcher and Vogtmann's Degree Theorem in Auter space An. There is a filtration of An into subspaces An,k using the degree of a graph, and the Degree Theorem says that each An,k is (k-1)-connected. This result is useful, for example to calculate stability bounds for the homology of Aut(Fn). 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.


The second named author gratefully acknowledges support from the SFB 701 of the DFG

Author information

Robert McEwen:
Ruckersville, VA 22968

Matthew C. B. Zaremsky:
Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902