Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory. Vassily Olegovich Manturov

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory

Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov

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“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 figure we denote by figure, that is, the alphabet figure1 consists of the letters from the alphabet figure, their inverses and the symbol “1”. Let Δ be a map and for each edge e of the map Δ a letter φ(e) ∈ figure1 is chosen (edges with φ(e) ≡ 1 are called 0-edges of the map; other edges are called figure–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 figure.

      Here the symbol “≡” denotes the graphical equality of the words in the alphabet figure. 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 figure 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 figure is a basis of a free group F = F(figure), figure is a set of words in the alphabet figure and there exists an epimorphism π : F(figure) → G such that its kernel is the normal closure of the subset {[r] | r ∈ figure} of the set of words F(figure). Elements of figure are called the relations of the presentation figurefigure|figurefigure. We will always suppose that every element r ∈ figure is a non-empty cyclically-irreducible word, that is, every element r of figure 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 figure.

      Definition 1.2. A cell of the diagram Δ is called a figure-cell if the label of its contour is graphically equal (up to cyclic permutations) either to a word R ∈ figure, 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 figure 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 figure-cell or a 0-cell.

      Earlier we gave two examples of diagrams used to show that a certain equality of the type W = 1 holds in a group given by its presentation. In fact this process is made possible by the following lemma due to van Kampen:

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