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Injective maps between Artin groups. (English) Zbl 1001.20034
Cossey, John (ed.) et al., Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14-19, 1996. Berlin: de Gruyter. 119-137 (1999).
Summary: A sufficient condition is given for the injectivity of a homomorphism between Artin monoids which moreover ensures injectivity of the induced map between Artin groups in the case where both groups are of finite type. We list numerous examples of monoid homomorphisms satisfying this injectivity condition, all of which happen to be so-called LCM-homomorphisms. In the case of an LCM-homomorphism there is a natural way to realise the corresponding map on Artin groups geometrically as the map induced on fundamental groups by an inclusion of certain finite simplicial complexes. An interesting group homomorphism which is not realised in this way exhibits the Artin group of type \(B_n\) as a subgroup of finite index of the classical \((n+1)\)-string braid group (type \(A_n\)). This subgroup is in fact the group of \(n\)-string braids over an annulus.
For the entire collection see [Zbl 0910.00040].

20F36 Braid groups; Artin groups
20E36 Automorphisms of infinite groups
20M05 Free semigroups, generators and relations, word problems
20M15 Mappings of semigroups