Symmetries of unimodal singularities and complex hyperbolic reflection groups

• Main Author: Haddley, Joel A.
• Other Authors: Goryunov, Victor V. ; Pratoussevitch, Anna
• Published: University of Liverpool 2011
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.542548

In search of discrete complex hyperbolic reflection groups in a singularity context, we consider cyclic symmetries of the 14 exceptional unimodal function singularities. In the 3-variable case, we classify all the symmetries for which the restriction of the intersection form of an invariant Milnor fibre to a character subspace has positive signature 1, and hence the corresponding equivariant monodromy is a reflection subgroup of U(k − 1,1). For such subspaces, we construct distinguished vanishing bases and their Dynkin diagrams. For k = 2, the projectivised hyperbolic monodromy is a triangle group of the Poincaré disk. For k = 3, we identify some of the projectivised monodromy groups within a recently published survey by J. R. Parker.