Dehn twists and free subgroups of symplectic mapping class groups

TL;DR

This paper investigates conditions under which Dehn twists about Lagrangian spheres generate free subgroups in symplectic mapping class groups, extending previous results and applying to Milnor fibers of certain singularities.

Contribution

It generalizes Ishida's result from Riemann surfaces to higher-dimensional symplectic manifolds and extends Seidel's exact sequence to arbitrary powers of Dehn twists.

Findings

Dehn twists about certain Lagrangian spheres generate free non-abelian subgroups.

Milnor fibers of isolated degenerate hypersurface singularities contain such pairs of spheres.

The categorical Seidel sequence is extended to arbitrary powers of Dehn twists.

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 proof generalises the categorical version of Seidel's long exact sequence 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.