POSITIVE TUPLES OF FLAGS, PIECEWISE CIRCULAR WAVEFRONTS, AND THE 3-DIMENSIONAL EINSTEIN UNIVERSE

Abstract

Fock and Goncharov defined the notion of positive subsets of a complete flag manifold G/B in order to study higher Teichmüller spaces. In this dissertation, we study the finite positive subsets when G = PSp(4,R) ∼= SO0(3,2). The main tool is the fact that the 3-dimensional Einstein universe, or Lie quadric, is one of the parabolic homogeneous spaces of G and it parametrizes oriented circles in the 2-sphere. We interpret complete flags in this setting as pointed oriented circles in the 2-sphere and the action of G as contactomorphisms of the unit tangent bundle of S2. This leads to an interpretation of positive subsets in G/B in terms of oriented piecewise circular curves in the 2-sphere, or equivalently piecewise linear Legendrian curves in RP3. We parametrize positive triples of flags by a pair of real-valued cross ratios. We explicitly describe a homeomorphism between the configurations space of positive triples of flags and the moduli space of 6-sided, labeled, positive, oriented piecewise circular wavefronts in S2.