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Artin groups of extra-large type are biautomatic. (English) Zbl 0872.20036
Summary: We develop new techniques to work with small cancellation theory diagrams for Artin groups. Using these techniques we examine paths in the Cayley graph of the Artin group. For any Artin group \(G\), with semigroup generators \(\mathcal A\), we define a language \(L(G)\subset{\mathcal A}^*\). The language \(L(G)\) is a set of canonical forms for the Artin group. In the case \(G\) is an Artin group of extra-large type or a two generator Artin group, we analyze the geometry of the small cancellation theory diagrams and show that \(L(G)\) is the language of a biautomatic structure for \(G\).

MSC:
20F36 Braid groups; Artin groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:
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