Whitehead graphs on handlebodies
John R. Stallings · 2012
A subset A of a free group F is called "separable" when there is a non-trivial free factorization F = F1 F2 such that each element of A is conjugate to an element of F1 or of F2. A single element is separable if and only if it belongs to a proper free factor. An algorithm is given to detect if a given nite set A is separable or not; this depends on cut vertices in the Whitehead graph of A relevant toagiven free basis X of F. Disjoint simple closed curves A on the boundary of a handlebody H are said to be "geometrically separable" when there is a disk D properly and non-trivially embedded in H whose boundary does not intersect any element of A. It is shown that separable in the algebraic sense implies geometrically separable.