The area theorem for the disc diagram over $C(3)$-$T(6)$-group

N. V. Bezverkhniy

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 18-27.

Résumé

Geometric methods are widely used in combinatorial group theory. The theory of small cancellation groups use the diagram method. In particular, it allows to approach various algorithmic problems. One of them is the power conjugacy problem. It is already solved for groups with a presentation satisfying the small cancellation conditions $C(3)$ and $T(6)$. However, it remains open for a similar class of groups, having a presentation satisfying the small cancellation conditions $C(6)$ and $T(3)$. In this paper we investigate the structure of connected diagrams over presentations satisfying the small cancellation conditions $C(3)$ and $T(6)$ and we indicate how our results may be possible used in the power conjugacy problem. The main result of this article is the proof of the theorem about lower bound on square of the reduced diagram on the group with small cancellation conditions $C(3)$-$T(6)$. It is known that for groups with conditions $C(p)$-$T(q)$ with $(p,q)\in \{(3,6), (4,4), (6,3)\}$, being automatic, isoperimetric inequality is quadratic. The same stated in well-known in small cancellation theory theorem of the square. Both statements restrict the area of the simply connected diagrams in the considered class of groups by the quadratic function of the length of the boundary. In this article it is proved that the lower bound for the area of the diagram of the specified type also is a quadratic function of the length of the border. The importance of this result is visible from the point of view of evaluation of complexity of the algorithm solves the word problem. It is not less than quadratic complexity of the length of the compared words. Bibliography: 15 titles.

Détail

Keywords: map, diagram, dual map, dehn region, band, ring diagram, small cancellation condition, defining relation, generators.

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