Dehn twists and free subgroups of symplectic mapping class groups

Abstract

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces [‘The structure of subgroup of mapping class groups generated by two Dehn twists’, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996) 240–241]. The proof generalizes the categorical version of Seidelˈs long exact sequence [‘A long exact sequence for symplectic Floer cohomology’, Topology 42 (2003) 1003–1063] to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.