Abstract: The main algorithmic problems in group theory formulated at the beginning of XX century, are the problem of words, the problem of the conjugation of words and the isomorphism problem for finitely presented groups. P. S. Novikov proved the unsolvability of the main algorithmic problems in the class of finitely presented groups. Therefore, algorithmic problems are studied in particular groups. In 1983, K. Appel and P. Schupp defined a class of Artin groups of extra-large type, in which they solved the problems of equality and conjugation of words. In 2003, V. N. Bezverkhnii introduced the class of Artin groups with a tree structure. In the graph corresponding to the Artin group, it is always possible to allocate the maximum subgraph corresponding to the Artin group with a tree structure. V. N. Bezverkhnii and O. Y. Platonova solved algorithmic problem in the class of Artin groups. The article examines the structure of diagrams over generalized tree structures of Artin groups, which are tree products of Artin groups of extra-large type and Artin groups with a tree structure, amalgamated by cyclic subgroups corresponding to the generators of these groups, and their application to the effective writing out generators of the centralizer of an element and solving the problem of conjugation of words in this class of groups. The proof of the main results uses the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii.
Keywords: Artin group, algorithmic problems, tree product of groups, diagram
For citation: Dobrynina, I. V. and Ugarov, A. S. On Generalized Tree Structures of Artin Groups, Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 52-63. DOI 10.46698/l4033-4336-3582-u
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