On the isomorphism problem for even Artin groups

Abstract

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)≃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)≃TGr(H).