#### 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
##### Abstract

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 ${F}_{2}$ 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 $1∕{\ell }^{2}$, where $\ell$ is the number of edges in the graph.

##### Keywords
free groups, primitive elements
Primary: 20E05
Secondary: 20F65