a geometric interpretation of group relations) is the first example of the main principle this book is devoted to: how to study different objects (in our case — groups) with geometric, “picture-valued” instruments; how to construct a picture which contains all the important information about the studied object.
1.1Group diagrams language
In this section we discuss diagrammatic language of group description. This approach was first discovered and used by van Kampen [van Kampen, 1933]. The essence of his discovery was an interconnection between combinatorially-topological and combinatorially-group-theoretical notions. The gist of this approach is a presentation of groups by flat diagrams (that is, geometrical objects, — flat complexes, — on a plane or other surfaces, such as a sphere or a torus). We review this theory following [Olshanskii, 1989].
1.1.1Preliminary examples
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
1.1.2The notion of a diagram of a group
Now we can move on to the explicit definitions of group diagrams and the overview of necessary results in that theory.
In accordance with [Olshanskii, 1989] in the present chapter a cell partitioning Δ of a surface S will be called a map on S for short. For some particular surfaces we will also use special names; for example, a map on a disc will be called a disc map, on an annulus an annular map, on a sphere or a torus — spherical or toric, respectively. Oriented sides of the partitioning are called edges of the map. Note that, if e is an edge of a map Δ, then e−1 is also its edge with the opposite orientation (consisting of the same points of the surface S as a side of the partitioning Δ).
Now consider an oriented surface S with a given map Δ and let us fix an orientation on its cells — e.g. let us walk around the boundary of each cell counterclockwise. In particular, the boundary of a disc map will be read clockwise and for an annular map, one boundary component (“exterior”) will be read clockwise, and another (“interior”) — counterclockwise.
Let a boundary component Y of a map or a cell consist of n sides. Walking around this component in accordance with the chosen orientation, we obtain a sequence of edges e1, . . . , en forming a loop. This loop will be called a contour of the map or the cell. In particular, a disc map has one contour, and an annular map has two contours (exterior and interior). Contours are considered up to a cyclic permutation, that is, every loop ei . . . ene1. . . ei−1 gives the same contour. A contour of a cell Π will be denoted by ∂Π and we will write e ∈ ∂Π if an edge e is a part of the contour ∂Π and we will call this situation “the edge e lies in the contour ∂Π”. Note that, even if an edge e lies in a contour ∂Π, its inverse e−1 does not necessarily lie in that contour. For example, in the situation depicted in Fig. 1.1 an edge a lies in the innermost triangular contour, but a−1 does not lie there.
Given a path p we may define a subpath in a natural way: a path q is a subpath of the path p if there exist two paths p1, p2 such that p = p1qp2. In the same way a subword is defined.
Given an alphabet
Definition 1.1. If for each edge
