Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 11

First, let us present several examples of the principle, which will be rigorously defined later in this section.

Example 1.1. Consider a group G with relations a³ = 1 and bab⁻¹ = c. Clearly in such group we have c³ = 1. This fact can be seen on the following diagram, see Fig. 1.1.

Fig. 1.1Diagrammatic view of the c³ = 1 relation

Indeed, if we go around the inner triangle of the diagram, we get the relation a³ = 1 (to be precise, the boundary of this cell gives the left-hand side of the relation; if we encounter an edge whose orientation is compatible with the direction of movement, we read the letter which the edge is decorated with, otherwise we read the inverse letter; in this example we fix the counterclockwise direction of movement). Similarly, the quadrilaterals glued to the triangle lead the second given relation c⁻¹bab⁻¹ = 1. Now if we look at the outer boundary of the diagram, we read c³ = 1, and that is what we need to prove.

This simple example gives us a glimpse of the general strategy: we produce a diagram, composed of cells, along the boundary of which given relations can be read. Then the outer boundary of the diagram gives us a new relation which is a consequence of the given ones.

Let us consider a bit more complex example of the same principle.

Example 1.2. Consider a group G where the relation x³ = 1 holds for every x ∈ G. It is a well-known theorem that in such a group every element a lies in some commutative normal subgroup N ⊂ G. Such situation arises, for example, in link-homotopy.

To prove this fact it is sufficient to prove that any element y = bab⁻¹ conjugate to a commutes with a. If that were the case, the subgroup N could be constructed as the one generated by all the conjugates of a.

So we need to prove that for every b ∈ G the following holds:

or, equivalently

This equality can be read walking clockwise around the outer boundary of the diagram in Fig. 1.2 composed of the relations b³ = 1, (ab)³ = 1, and (a⁻¹b)³ = 1.

Fig. 1.2Proof of the claim given in Example 1.2