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

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Invariants And Pictures: Low-dimensional Topology And Combinatorial Group Theory - Vassily Olegovich Manturov Series On Knots And Everything

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graphically equal to W.

      To prove the inverse implication, consider an unoriented diagram figure with the label of its contour figure. We need to show that for each resolution W of the word figure the relation W = 1 holds in the group G.

      First, note that if this relation holds for one resolution of the word figure, it holds for every other resolution of this word. Indeed, due to the relations a2 = 1 we may freely replace letters with their inverses: the relation uav = ua−1v holds in the group G for any subwords u, v and any letter a.

      But by definition the diagram figure is obtained from some diagram Δ over the group G by forgetting the orientation of its edges. Therefore there exists a van Kampen diagram over the group G with the label of its contour graphically equal to some resolution R of the word figure Therefore due to the van Kampen lemma R = 1 in the group G. And thus for every other resolution W of the word figure the relation W = 1 holds in the group G.

       1.2.1Small cancellation conditions

      We will introduce the notion of small cancellation conditions C′(λ), C(p) and T(q). Roughly speaking, the conditions C′(λ) and C(p) mean that if one takes a free product of two relations, one gets “not too many” cancellations. To give exact definition of those objects, we need to define symmetrisation and a piece.

      As before, let G be a group with a presentation (1.1). Given a set of relations figure its symmetrisation figure* is a set of all cyclic permutations of the relations r ∈ figure and their inverses. A word u is called a piece with respect to figure if there are two distinct elements w1, w2 ∈ figure* with the common beginning u, that is w1 = uv′, w2 = uv″. The length of a word w (the number of letters in it) will be denoted by |w|.

      Definition 1.7. Let λ be a positive real number. A set of relations figure is said to satisfy the small cancellation condition C′(λ) if

      for every r ∈ figure* and its any beginning u which is a piece with respect to figure.

      Definition 1.8. Let p be a natural number. A set of relations figure is said to satisfy the small cancellation condition C(p) if every element of figure* is a product of at least p pieces.

      The small cancellation conditions are given as conditions on the set of relations figure. If a group Γ admits a presentation figureS|figurefigure with the set of relations satisfying the small cancellation condition, then the group Γ is said to satisfy this condition as well.

      The condition C′(λ) is sometimes called metric and the condition C(p) — non-metric. Note that figure always yields C(n + 1).

      There exists one more small cancellation condition. Usually it is used together with either of the conditions C′(λ) or C(p).

      Definition 1.9. Let q be a natural number, q > 2. A set of relations figure is said to satisfy the small cancellation condition T(q) if for every l ∈ {3, 4, . . . , q − 1} and every sequence {r1, r2, . . . , rl} of the elements of figure* the following holds: if figure, then at least one of the products r1r2, . . . , rl−1rl, rlr1 is freely reduced.

      Remark 1.1. Those conditions have a very natural geometric interpretation in terms of van Kampen diagrams. Namely, the C(p) condition means that every interior cell of the corresponding disc partitioning has at least p sides; the T(q) condition means that every interior vertex of the partitioning has the degree of at least q.

      Note that every set figure satisfies the condition T(3). Indeed, no interior vertex of a van Kampen diagram has degree 1 or 2.

      An important problem of combinatorial group theory is the word problem: the question whether for a given word W in a group G holds the equality W =1 (or, more generally, whether two given words are equal in a given group). Usually the difficult question is to construct a group where a certain set of relations holds but a given word is nontrivial. In other words, to prove that a set of relations figure does not yield W = 1 (note that there may exist groups where both the relations figure and 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.

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