On the isomorphism problem for even Artin groups

Rubén Blasco-García (1) , Luis Paris (2)

1 University of Zaragoza - Universidad de Zaragoza [Zaragoza]
2 IMB - Institut de Mathématiques de Bourgogne [Dijon]

An even Artin group is a group which has a presentation with relations of the form (st)(n) = (ts)(n) with n >= 1. With a group G we associate a Lie Z-algebra TGr(G). This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group G we determine a presentation for TGr(G). By means of this presentation we obtain information about the diagram of G. We then prove an isomorphism criterion for Coxeter matrices that ensures that the diagram of G is uniquely determined by this information. Let d >= 2. We show that, if two even Artin groups G and H having presentations with relations of the form (st)(dk) = (ts)(dk) with k >= 0 are such that TGr(G) similar or equal to TGr(H), then G and H have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups G and H such that TGr(G) similar or equal to TGr(H).

Mots clés: Artin groups; Isomorphism problem

Domaines: Mathématiques [math]

Rubén Blasco-García, Luis Paris. On the isomorphism problem for even Artin groups. Journal of Algebra, 2022, 607, pp.35-52. ⟨10.1016/j.jalgebra.2020.05.025⟩. ⟨hal-03895913⟩