ON THE ROOT PROBLEM IN SOME CONSTRUCTIONS OF ARTIN GROUPS Abstract (English):
An algorithm for solving the root problem in a tree product of Artin groups with a tree structure and extra-large type is considered.

Keywords:
Artin group, tree structure, extra-large type, root problem, diagram

Consider a group    – a word from  sequentially taken generators  ,  – numbers that are elements of the Coxeter matrix .

At  for arbitrary, unequal indices, we have an extra large type  .

If the graph in the image  is a tree, then we obtain a tree structure .

Consider further the following construction:  which is the product of G_k having a tree structure or extra-large type, where  – is the degree of the generator  ,  – is the degree of the generator  .

In the factors of this construction, the problems of identity (equality) and conjugacy are solvable -.

Consider diagrams over a construction  . The conjugacy of elements will be denoted by the symbol  ̴ .

Theorem 1. The R-diagram of the conjugacy of non-powers of generators of cyclically R,  – irreducible elements of  is one-layer.

The R-diagram of the equality of non-powers of the generators of  R,  -irreducible elements of  is one-layer.

The proof of the theorem is similar to the proof presented in .

The R-diagram of the conjugacy of elements that are cyclically R and  -irreducible is called especially special  when there is a single region for which the syllable length of the intersection mark with one of the boundaries by 2 units differs from the syllabic length of the intersection mark with the other of the boundaries. We get that on one of the boundaries the syllable length of the element is less than on the other. Then the replacement of an element with a shorter syllable length is called a special annular contraction .

Definition 2 . An element is called dead-end if the special annular cancellation is not applicable to it.

It is possible to construct an algorithm that establishes for any element whether it will be a dead end, which follows from  - .

Lemma 2. If two dead-end elements are conjugate, then their syllable lengths are equal.

Indeed, if one of the dead-end elements has a large length and is conjugated to a given one, then the abbreviations defined above can be performed in it. This can be easily seen in the contingency diagram of the elements.

Lemma 3. The construction  is torsion-free.

The proof follows from the fact that if an element of G is of finite order, then it must be conjugate to an element of finite order of some factor, but it follows from ,  that the factors are torsion-free. Consequently, it is absent in the construction under consideration.

Lemma 4 . Given a cyclically R,  -irreducible nontrivial element u, then there exists an element v, for which every power of  R,  -irreducible, and v ̴ u, or v ̴ u2.

Definition 3 . Problem 1 is understood as the problem of constructing an algorithm that determines whether one element is a nonunit power of some other element.

Theorem 2. In the construction of G, Problem 1 is algorithmically solvable.

Lemma 5 . Problem 1 is solvable in tree-structure factors of the construction G.

Lemma 6 . Problem 1 is solvable in factors of construction G of extra-large type.

Let us proceed to the proof of the theorem.

First of all, note that Problem 1 is solvable in the factors of G on the basis of Lemmas 5 and 6. Therefore, we will consider the general case when there are at least two factors.

Suppose that in the construction G under consideration for some elements  the equality  (1)

Let us square both sides of equality (1). We get  .                                                              (2)

Carrying out the necessary cancellations in u  we replace it with an element    in accordance with Lemma 4 so that any degree  R,  -irreducible. Then equality (2) takes the form:  (3)

The element  is equal to some R,  -irreducible element