Vol. 7, No. 4, 2014

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Whitehead graphs and separability in rank two

Matt Clay, John Conant and Nivetha Ramasubramanian

Vol. 7 (2014), No. 4, 431–452

By applying an algorithm of Stallings regarding separability of elements in a free group, we give an alternative approach to that of Osborne and Zieschang in describing all primitive elements in the free group of rank 2. As a result, we give a proof of a classical result of Nielsen, used by Osborne and Zieschang in their work, that the only automorphisms of F2 that act trivially on the abelianization are those defined by conjugation. Finally, we compute the probability that a Whitehead graph in rank 2 contains a cut vertex. We show that this probability is approximately 12, where is the number of edges in the graph.

free groups, primitive elements
Mathematical Subject Classification 2010
Primary: 20E05
Secondary: 20F65
Received: 14 March 2012
Accepted: 27 December 2012
Published: 31 May 2014

Communicated by Gaven Martin
Matt Clay
Department of Mathematical Sciences
University of Arkansas
SCEN 301
Fayetteville, AR 72701
United States
John Conant
Department of Education
Shepherd University
Shepherdstown, WV 25443
United States
Nivetha Ramasubramanian
Clinton Township, MI 48036
United States