Whitehead Graphs and Separability

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This Demonstration draws the Whitehead graph of a reduced word in a free group of given rank. The word is first cyclically reduced. Multiple edges between vertices are indicated with numbered labels. If the Whitehead graph is connected and does not have a cut vertex, then the word is not separable. Otherwise, by pressing the "forward" button, a Whitehead automorphism is applied to reduce the number of edges.

Contributed by: Matt Clay (March 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The Whitehead graph of an element in a free group relative to the basis is a graph with vertices labeled with an edge between and for each two-letter subword of considered as a cyclic word in the given basis. For example, the subword would correspond to an edge between and and the subword would correspond to an edge between and . Algebraic properties of are reflected in the structure of the graph. One such property is whether or not there exists a basis for the free group such that is conjugate to a word omitting one of the generators . In terms of the Whitehead graph, this means there is a pair of vertices not adjacent to any edge. If so, is called separable. Stallings showed that if is separable, then every Whitehead graph of is either disconnected or has a cut vertex, that is, a vertex that when removed results in a disconnected graph. A cut vertex of a connected graph is also referred to as an articulation vertex.

The algorithm to detect if is separable is as follows. If the Whitehead graph of does not have an isolated vertex and is disconnected, then there is a certain operation , called a Whitehead automorphism, such that the Whitehead graph of has a pair vertices not adjacent to any edges, and hence is separable. Then using the basis we can observe that is separable. Namely, when written in this new basis, the generator is omitted.

Next, we check for a cut vertex. If the Whitehead graph does not have a cut vertex, then is not separable. Otherwise, the Whitehead graph has a cut vertex. In this case, there is a Whitehead automorphism such that the Whitehead graph for has fewer edges. Go back to the beginning of the algorithm using this new graph. As the number of edges decreases whenever this last possibility occurs, the process will eventually terminate. By keeping track of the Whitehead automorphisms that arise, we can find the basis that witnesses as separable if indeed it is.

The snapshots depict using the algorithm to show that the element in the free group of rank two is separable.

Snapshot 1: The Whitehead graph of does not have an isolated vertex but is connected and contains a cut vertex. Specifically, the vertex labeled (The vertex labeled is also a cut vertex.) The Whitehead automorphism that will reduce the number of edges is displayed.

Snapshot 2: After pressing the "forward" button, the Whitehead automorphism is applied and the resulting graph is now disconnected. The effect of on the basis is shown on the bottom of the panel. The original word is written (cyclically) as in the basis and .

Snapshot 3: After pressing the "forward" button again, a Whitehead automorphism is applied. The resulting graph has an isolated vertex. This shows that is separable. The original word is written (cyclically) as in the basis and .

Inverses of basis elements are denoted by capital letters: , etc.

Reference

[1] J. R. Stallings, "Whitehead Graphs in Handlebodies," in Geometric Group Theory Down Under (J. Cossey and C. F. Miller, eds.), Berlin, Germany: Walter de Gruyter, 1996 pp. 317–330.



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