Action of Braid Groups on Determinantal Ideals, Compact Spaces and a Stratification of Teichmüller Space
نویسندگان
چکیده
We show that for n ≥ 3 there is an action of the braid group Bn 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 Bn on certain compact subspaces of some Euclidean spaces of dimension n 2 . These subspaces are real semi-algebraic varieties and include spheres of dimension n 2 −1 on which the kernel of the action of Bn is the centre of Bn. We investigate the action of Bn on these subspaces. We also show how a finite number of disjoint copies of the Teichmüller space for the n-punctured disc is naturally a subset of this R n 2 and how this cover (in the broad sense) of Teichmüller space is a union of non-trivial Bn-invariant subspaces. The action of Bn on this cover of Teichmüller space is via polynomial automorphisms. For the case n = 3 we show how to define modular forms on the 3-dimensional Teichmüller space relative to the action of B3. §
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