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

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

W = 1 hold due to the presence of additional relations). Small cancellation theory proves to be a very powerful and useful instrument in that situation. In particular, an important role in solution of that kind of problems plays the Greendlinger theorem which we will formulate in this section. We will always assume that the presentation (1.1) is symmetrised.

The Greendlinger theorem deals with the length of the common part of cells’ boundary. Sometimes two cells are separated by 0-cells. Naturally, we should ignore those 0-cells. To formulate that accurately we need some preliminary definitions.

Let Δ be a reduced diagram over a symmetrised presentation (1.1). Two -edges e₁, e₂ are called immediately close in Δ if either e₁ = e₂ or e₁ and (or and e₂) belong to the contour of some 0-cell of the diagram Δ. Furthermore, two edges e and f are called close if there exists a sequence e = e₁, e₂, . . . , el = f such that for every i = 1, . . . , l − 1 the edges ei, e_i+1 are immediately close.

Now for two cells Π₁, Π₂ a subpath p₁ of the contour of the cell Π₁ is a boundary arc between Π₁ and Π₂ if there exists a subpath p₂ of the contour of the cell Π₂ such that p₁ = e₁u₁e₂ . . . u_n−1en, , the paths ui, vj consist of 0-edges, ei, fi are -edges such that for each i = 1, . . . , n the edge fi is close to the edge ei. In the same way a boundary arc between a cell and the contour of the diagram Δ is defined.

Informally we can explain this notion in the following way. Intuitively, a boundary arc between two cells is the common part of the boundaries of those cells. The boundary arc defined above becomes exactly that if we collapse all 0-cells between the cells Π₁ and Π₂.

Now consider a maximal boundary arc, that is a boundary arc which does not lie in a longer boundary arc. It is called interior if it is a boundary arc between two cells, and exterior if it is a boundary arc between a cell and the contour ∂Δ.

Remark 1.2. It is easy to see that the small cancellation conditions C′(λ) and C(p) have a natural geometric interpretation in terms of boundary arcs. Namely, an interior arc of a cell Π of a reduced diagram of a group satisfying the C′(λ) condition has length smaller than λ|∂Π|. Likewise, the C(p) condition means that the boundary of every cell of the corresponding diagram consists of at least p arcs.

Now we can formulate the Greendlinger theorem.

Theorem 1.3. Let Δ be a reduced disc diagram over a presentation of a group G satisfying the small cancellation condition C′(λ) for some and let Δ have at least one -cell. Further, let the label φ(q) of the contour q = ∂Δ be cyclically irreducible and such that the cyclic word φ(q) does not contain any proper subwords equal to 1 in the group G.

Then there exists an exterior arc p ofsome -cell Π such that

Remark 1.3. If we formulate the Greendlinger theorem for unoriented diagrams, the theorem still holds.

Before proving this theorem, let us interpret it in terms of group presentation and the word problem. Consider a group G with a presentation (1.1) satisfying the small cancellation condition C′(λ), . Due to van Kampen lemma 1.1 (and its strengthening, Theorem 1.1) for a word W = 1 in the group G there exists a diagram with boundary label W. Boundary label of every -cell of the diagram by definition is some relation from the set (or its cyclic permutation). Then, due to Greendlinger Theorem 1.3 there is an -cell such that at least half of its boundary “can be found” in the boundary of the diagram. Thus we obtain the following corollary (sometimes it is also called the Greendlinger theorem):

Theorem 1.4 (Greendlinger [Greendlinger, 1961]).

Let G be a group with a presentation (1.1) satisfying the small cancellation condition C′(λ), . Let W ∈ F be a nontrivial freely reduced word such that W = 1 in the group G. Then there exist a subword V of W and a relation R ∈ such that V is also a subword of R and

This theorem is very useful in solving the word problem.

Example 1.3 (A.A. Klyachko). Consider the relation R = [x, y]² and the word

The question is, whether in every group with the relation R the equality W = 1 holds.

First, it is easy to see that the set of relations _* obtained from R by symmetrisation satisfies the condition . Therefore, due to the Greendlinger theorem every word V such that V = 1 in the group G has a cyclic permutation such that both its irreducible form and some relation have a common subword p of length .

On the other hand, the longest common subwords of the word W = [x¹⁰⁰⁰, y¹⁰⁰⁰]¹⁰⁰⁰ and any of the relations have length 2: those are

Therefore we can state that there exists a group G where for some two elements a, b ∈ G [a, b]² = 1 but [a¹⁰⁰⁰, b¹⁰⁰⁰]¹⁰⁰⁰ ≠ 1.

Now let us prove Theorem

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