Action of braid groups on determinantal ideals, compact spaces and a stratification of Teichmuller space

摘要

We show that for n ≥ 3 there is an action of the braid group B_n on the determinantal ideals of a certain n * n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of B_n on certain compact subspaces of some Euclidean spaces of dimension (_2~n). These subspaces are real semi-algebraic varieties and include spheres of dimension (_2~n) - 1 on which the kernel of the action of B_n is the centre of B_n. We investigate the action of B_n on these subspaces. We also show how a finite number of disjoint copies of the Teichmuller space for the n-punctured disc is naturally a subset of this R~((_2~n)) and how this cover (in the broad sense) of Teichmuller space is union of non-trivial B_n-invariant subspaces. The action of B_n on this cover of Teichmuller space is via polynomial automorphisms. For the case n = 3 we show how to define modular forms on the 3-dimensional Teichmuller space relative to the action of B_3.