Artin groups and geometric monodromy
The main result of this paper says that Artin groups (generalized braid groups) of type \(E_6\), \(E_7\), \(E_8\) cannot be embedded into the mapping class group of a surface in such a way that the standard generators of the Artin group are mapped onto Dehn twists. The groups \(E_i\) are the domains of geometric monodromy homomorphisms of isolated singularities of the same name, where the target group is the mapping class group of a surface. The monodromy homomorphism takes the standard generators of the Artin group onto Dehn twists. The above result gives a negative answer to a question of Sullivan: is the geometric monodromy of an isolated singularity of a curve always injective? Theorem 3. If \(\Gamma\) is a graph of type \(E_6\) or \(E_7\) or \(E_8\) then there is no geometric embedding of the corresponding Artin group \(A\) into the mapping class group of a surface. -- Corollary 5.1. If \(A\) is an Artin group with finite exponent which corresponds to a connected Dynkin diagram and admits a geometric embedding into the mapping class group of a surface then \(A\) is a group of type \(A_n\) or \(D_n\). -- Theorem 6. The geometric monodromy of a plane curve singularity is not injective for simple singularities of type \(E_6\), \(E_7\) and \(E_8\).
scientific article; zbMATH DE number 1383861