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

      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.

1.1.3The van Kampen lemma

      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:

Lemma 1.1 (van Kampen [van Kampen, 1933]). Let W be an arbitrary non-empty word in the alphabet ¹. Then W = 1 in a group G given by its presentation (1.1) if and only if there exists a disc diagram over the presentation (1.1) such that the label of its contour graphically equals W.

      Proof. 1) First, let us prove that if Δ is a disc diagram over the presentation (1.1) with contour p, its label φ(p) = 1 in the group G.

      If the diagram Δ contains exactly one cell Π, then in the free group F we have either φ(p) = 1 (if Π is a 0-cell) or φ(p) = R^±1 for some R ∈ (if Π is an -cell). In any case, φ(p) = 1 in the group G.

      If Δ has more than one cell, then the diagram can be cut by a path t into two disc diagrams Δ₁, Δ₂ with fewer cells. We can assume that their contours are p₁t and p₂t⁻¹ where p₁p₂ = p. By induction it holds that φ(p₁t) = 1 and φ(p₂t⁻¹) = 1 in the group G. Therefore

      in the group

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