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Directed diagrammatic reducibility. (English) Zbl 07285174
Summary: We introduce the notion of directed diagrammatic reducibility which is a relative version of diagrammatic reducibility. Directed diagrammatic reducibility has strong group theoretic and topological consequences. A multi-relator version of the Freiheitssatz in the presence of directed diagrammatic reducibility is given. Results concerning asphericity and $$\pi_1$$-injectivity of subcomplexes are shown. We generalize the Corson-Trace characterization of diagrammatic reducibility to directed diagrammatic reducibility. We compare diagrammatic reducibility of relative presentations to directed diagrammatic reducibility. Classical tools for showing diagrammatic reducibility, such as the weight test, the max/min test, and small cancellation techniques are adapted to directed diagrammatic reducibility. The paper ends with some applications to labeled oriented trees.
##### MSC:
 57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) 57M05 Fundamental group, presentations, free differential calculus 20F06 Cancellation theory of groups; application of van Kampen diagrams
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