Читать книгу Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov - Страница 12

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. . . e_i−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 we denote by , that is, the alphabet ¹ consists of the letters from the alphabet , their inverses and the symbol “1”. Let Δ be a map and for each edge e of the map Δ a letter φ(e) ∈ ¹ is chosen (edges with φ(e) ≡ 1 are called 0-edges of the map; other edges are called –edges).

Definition 1.1. If for each edge e of a map Δ the following relation holds:

then the map Δ is called a diagram over .

Here the symbol “≡” denotes the graphical equality of the words in the alphabet . In other words the notation V ≡ W means that the words V and W are the same as sequences of letters of the alphabet. By definition we set 1⁻¹ ≡ 1.

When p = e₁. . . en is a path in a diagram Δ over let us define its label by the word φ(p) = φ(e₁). . . φ(en). If the path is empty, that is |p| = 0, then we set φ(p) ≡ 1 by definition. As before, a label of a contour is defined up to a cyclic permutation (and thus forms a cyclic word).

Consider a group G with presentation

That means that is a basis of a free group F = F(), is a set of words in the alphabet and there exists an epimorphism π : F() → G such that its kernel is the normal closure of the subset {[r] | r ∈ } of the set of words F(). Elements of are called the relations of the presentation |. We will always suppose that every element r ∈ is a non-empty cyclically-irreducible word, that is, every element r of or any of its cyclic permutations do not include subwords of the form ss⁻¹ or s⁻¹s for some s ∈ F.

Note that if a presentation of a group has a relation R, then it has all its cyclic permutations as relations as well.

Let Δ be a map over the alphabet .

Definition 1.2. A cell of the diagram Δ is called a -cell if the label of its contour is graphically equal (up to cyclic permutations) either to a word R ∈ , or its inverse R⁻¹, or to a word, obtained from R or from R⁻¹ by inserting several symbols “1” between its letters.

This definition effectively means that choosing direction and the starting point of reading the label of the boundary of any cell of the map and ignoring all trivial labels (the ones with φ(e) ≡ 1) we can read exactly the words from the set of relations of the group G and nothing else.

Sometimes it proves useful to consider cell with effectively trivial labels. To be precise, we give the following definition.

Definition 1.3. A cell Π of a map Δ is called a 0-cell if the label W of its contour e₁. . . en graphically equals φ(e₁). . . φ(en), where either φ(ei) ≡ 1 for each i = 1, . . . , n, or for some two indices i ≠ j the following holds:

and

Finally, we can define a diagram of a group.

Definition 1.4. Let G be the group given by a presentation (1.1). A diagram Δ on a surface S over the alphabet is called a diagram on a surface S over the presentation (1.1) (or a diagram over the group G for short) if every cell of this map is either an -cell or a 0-cell.