The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group
By J. Crisp, Luis Paris. 2000
89 citations recorded by the source corpus.
Abstract.Let A be an Artin group with standard generating set {σs:s∈S}. Tits conjectured that the only relations in A amongst the squares of the generators are consequences of the obvious ones, namely that σs2 and σt2 commute whenever σs and σt commute, for s,t∈S. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number ms≥2 for each s∈S, we show that the elements {Ts=σsms:s∈S} generate a subgroup that has a finite presentation in which the only defining relations are that Ts and Tt commute if σs and σt commute.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/400c01624f4ed5beb803956ef0c17110d657f862
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In Discrete Mathematics and Combinatorics
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Citations in the corpus
- On groups generated by two positive multi-twists: Teichmuller curves and Lehmer's number (2003, background): “For more on this see eg Crisp and Paris [14], Wajnryb [59], [60], Perron and Vannier [49], and A’Campo [1].”
- Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups (2010, background): “Right-angled Artin subgroups of mapping class groups have been studied by various authors, for example by Crisp and Paris in [15].”
- Braids: A Survey (2004, unknown): “Theorem 19 [49] Let S be a finite set, let Γ be a Coxeter graph over S, and let A(Γ) be the associated Artin group, as defined in §5.”
- An introduction to right-angled Artin groups (2006, unknown): “It was conjectured by Tits and proved by Crisp and Paris [33] that for any Artin group A, the subgroup generated by the squares of the generators, s(2)1, .”
- Artin monoids inject in their groups (2001, background): “The homomorphism ` is obtained by a \folding" as described in [Cri], its construction is essentially the same as the one given in [ CrP , Sec.”