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