Geometric monodromy groups of simple singularities (Q1817303)

Let \(f: (\mathbb{C}^2,0) \to \mathbb{C}\) be a holomorphic map with isolated singularity at 0 and \[ G= f+ \sum^{\mu - 1}_{i = 0} t_ig_i \] its universal unfolding. Let \(D(\eta) = \{z\in \mathbb{C}^\mu \mid |t |\leq \eta\}\), \(V= \{(x,y,t) \in B_\varepsilon \times D (\eta) \mid G(x,y,t) = 0\}\), \(\pi: V \to D (\eta)\) the projection, \(\sum\) the discriminate and \(F\) the Milnor fibre. The representation of the monodromy group \[ \xi: \pi_1 \bigl(D (\eta) \backslash \Sigma, t_0 \bigr) \to \pi_0 \bigl(\text{Diff}^+ (F, \partial F) \bigr) \] is studied. It is proved that it is faithful for the simple singularities \(A_n\) and \(D_n\).