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A generalized weight test with applications to tree presentations. (Ein verallgemeinerter Gewichtstest mit Anwendungen auf Baumpräsentationen.) (German) Zbl 0761.57003
We introduce the “cycle test”, which is a generalization of Gersten’s “weight test”, to study diagrammatic reducibility of standard 2- complexes. Diagrammatic reducibility (DR) is a stronger combinatorial form of asphericity for 2-complexes. A homogeneous version of the cycle test leads to conditions that generalize small cancellation theory in its application as a test for DR; a fact which also has been observed by Gersten. Aside from that, the cycle test has many applications in a nonhomogeneous way, of which we present several examples. In particular, some of these examples are related to “labelled oriented tree” presentations whose 2-complexes are spines of ribbon disk complements.
Reviewer: G.Huck

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)
20F06 Cancellation theory of groups; application of van Kampen diagrams
57M15 Relations of low-dimensional topology with graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
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