To achieve it we need for a given word W such that W = 1 in the group G to construct a diagram Δ with contour p such that φ(p) = W.
It is well-known that in the free group F the word W equals a word
Construct a polygonal line t1 on a plane and mark its segments with letters so that the line reads the word X1. Connect a circle s1 to the end of this line and mark it so that it reads
Construct the second diagram analogously for the word
Continue the process until we obtain a diagram Δ′ such that φ(∂Δ′) ≡ V, see Fig. 1.3.
Finally, gluing some 0-cells to the diagram Δ′ we can transform the word V into the word W getting a diagram Δ such that φ(∂Δ) ≡ W. That completes the proof.
This lemma means that disc diagrams can be used to describe the words in a group which are equal to the neutral element of the group. It turns out that annular diagrams can be used in a similar manner.
Fig. 1.3The diagram Δ′ with boundary label φ(∂Δ′) ≡ V
Lemma 1.2 (Schupp [Schupp, 1968]). Let V, W be two arbitrary nonempty words in the alphabet
Let p be a loop on a surface S such that its edges form a boundary of some subspace
A subdiagram of a given diagram Δ is a submap Γ of the map Δ with edges endowed with the same labels as in the map Δ. Informally speaking, a subdiagram is a disc diagram cut out from a diagram Δ.
Let us state an additional important result about the group diagrams.
Lemma 1.3. Let p and q be two (combinatorially) homotopic paths in a given diagram Δ over a presentation (1.1) of a group G. Then φ(p) = φ(q) in the group G.
In the next section the diagrammatic approach will be used to deal with groups satisfying the small cancellation conditions. In that theory a process of cancelling out pairs of cells of a diagram is useful (in addition to the usual process of cancelling out pairs of letters a and a−1 in a word). The problem is that two cells which are subject to cancellation do not always form a disc submap, so to define the cancellation process correctly we need to prepare the map prior to cancelling a suitable pair of cells. Let us define those notions in detail.
First, for a given cell partitioning Δ we define its elementary transformations (note that elementary transformations are defined for any cell partitioning, not necessarily diagram).
Definition 1.5. The following three procedures are called the elementary transformations of a cell partitioning Δ:
(1)If the degree of a vertex o of Δ equals 2 and this vertex is boundary for two different sides e1, e2, delete the vertex o and replace the sides e1, e2 by a single side e = e1 ∪ e2;
(2)If the degree of a vertex o of a n–cell Π (n ≥ 3) equals 1 and this vertex is boundary for a side e, delete the side e and the vertex o (the second boundary vertex of the side e persists);
(3)If two different cells Π1 and Π2 have a common side e, delete the side e (leaving its boundary vertices), naturally replacing the cells Π1 and Π2 by a new cell Π = Π1 ∪ Π2.
Now we can define a 0-fragmentation of a diagram Δ. First, consider a diagram Δ′ obtained form the diagram Δ via a single elementary transformation. This transformation is called an elementary 0-fragmentation if one of the following holds:
(1)The elementary transformation is of type 1 and either φ(e1) ≡ φ(e), φ(e2) ≡ 1 or φ(e2) ≡ φ(e), φ(e1) ≡ 1 and all other labels are left unchanged;
(2)The elementary transformation is of type 2 and φ(e) ≡ 1;
(3)The elementary transformation is of type 3 and one of the cells Π1, Π2 became a 0-cell.
Definition 1.6. A diagram Δ′ is a 0-fragmentation of a diagram Δ if it is obtained from the diagram Δ by a sequence of elementary 0-fragmentations.
Note that 0-fragmentation does not change the number of
Now consider an oriented diagram over a presentation (1.1). Let there be two
