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

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

Let G be a group with a presentation (1.1) and Δ be a diagram as in the statement of the theorem. Since Δ is a disc diagram, we can place it on the sphere. We construct the dual graph Φ to the 1-skeleton of the diagram Δ in the following way.

Place a vertex of the new graph Φ into each -cell of the diagram Δ and one vertex O into the outer region on the sphere. To construct the edges of the graph Φ (note that they are not edges of a diagram) we perform the following procedure. For every interior boundary arc between cells Π₁ and Π₂ we chose an arbitrary -edge e₁ and an edge e₂ close to it such that lies in ∂Π₂. Now we connect the vertices of the graph Φ, lying inside the cells Π₁, Π₂ by a smooth curve γ transversally intersecting the interior of the edges e₁ and e₂ once and crossing all 0-cells lying between them. This curve γ is now considered as an edge of the graph Φ. This procedure is performed for every vertex of the graph Φ and every boundary arc (if the arc is exterior, we connect the vertex with the “exterior” vertex O).

The main idea of the proof is to study the dual graph Φ and to prove the necessary inequality by the reasoning of Euler characteristic of sphere.

First, note that among the faces of the graph Φ there are no 1-gons (because the boundary labels of the cells of the diagram Δ are cyclically irreducible) or 2-gons (because in the previous paragraphs we have constructed exactly one edge crossing every maximal boundary arc of Δ). Therefore every face of Φ is at least a triangle. Thus, denoting the number of vertices, edges and faces of the graph Φ by V, E and F respectively, and considering the usual Euler formula V − E + F = 2 we get the following inequality:

Now, suppose that the statement of the theorem does not hold. For each edge of the graph Φ connecting the vertices oi, oj lying in the cells Π_i, Π_j of the diagram Δ we attribute this edge to each of those vertices with the coefficient ; if an edge connects a vertex ok with the vertex O, we attribute it to the vertex ok with the coefficient 1.

Fix an arbitrary vertex o ≠ O. Denote the cell where the vertex o lies by Π and consider the following possibilities.

(1)Let the contour ∂Π consist only of interior arcs. Due to the condition the length of each of those arcs is smaller than (see Remark 1.2). Therefore, their number is not smaller than seven and thus at least edges of the graph Φ is attributed to the vertex o.

(2)Let there be exactly one exterior arc p. Since we suppose that |p| ≤ |∂Π|, the number of interior arcs of the contour of the cell Π is at least four. Therefore we attribute at least edges to the vertex o.

(3)Let the cell Π have two exterior edges p₁, p₂. Note that the end of the arc p₁ cannot be the beginning of the arc p₂ (and vice versa) and they cannot be separated by 0-edges only because otherwise we could cut the diagram Δ with edges f₁, . . . , fl such that φ(fi) = 1 for all i = 1, . . . , l and due to van Kampen lemma obtain two proper subwords φ(q₁), φ(q₂) of the word φ(q) (where q is the contour of the diagram Δ) equal to 1 in the group G. That contradicts the condition of φ(q) being cyclically irreducible.

Therefore, the cell Π has at least two distinct interior boundary arcs and there are at least edges attributed to the vertex o.

(4)If the cell Π has at least three exterior arcs, there are at least three edges of the graph Φ attributed to the vertex o.

Considering those possibilities for all vertices of the graph Φ except the “exterior” vertex O we see that

and that contradicts the inequality (1.3). This contradiction completes the proof.

1.3Algorithmic problems and the Dehn algorithm

In this section we give a brief overview of two algorithmic problems in the combinatorial group theory which were already mentioned in the context of group diagrams — the word problem and the conjugacy problem.

Consider a group G given by its presentation (1.1) which is effective in the sense that for every word R we can algorithmically determine whether it lies in the set or not. There are two fundamental problems in the combinatorial group theory:

(1)The word problem: is there an algorithm which for any pair of words U, V in the group alphabet determines whether those words are equal in the group G or not?

(2)The conjugacy problem: is there an algorithm which for any pair of words U, V in the group alphabet determines whether those words are conjugate in the group G or not?

Naturally the word problem can be reformulated in the form

“is there an algorithm which for any word W in the group alphabet determines whether W = 1 is in the group G or not?”

because U = V is in the group G if and only if UV⁻¹ = 1 is in the group G.

It turns out that the small cancellation theory is a very useful instrument in the study of the word and conjugacy problems. In particular, the following algorithm (originally due to Dehn, see [Schupp, 1968]) solves the word problem in the groups with the condition C′(λ), .

The Dehn algorithm. The algorithm is inductive, so we may assume that for any word V of the length less than n we can algorithmically determine whether V = 1 is in the group G or not.

Now let W be a word of length n. We may assume that W is cyclically irreducible: for any proper subword of the cyclic word W we may determine by induction whether it equals 1 in the group G or not. If such a subword exists, replacing it with 1 we reduce the length of the word W and by induction solve the word problem for W.

So consider a cyclically irreducible word W, |W| = n. By the Greendlinger theorem if W = 1 in the group G, then the cyclic word W has a subword X such that there exists a relation R = XY ∈ Скачать книгу