Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

This chapter extends Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The resulting dynamical system is a continuous version of the action of the mapping class group of S on the deformation space. It observes how ergodic properties of this action relate to this flow. When G is compact, this flow is strongly mixing over each component of the deformation space and of each stratum of the Teichmüller unit sphere bundle over the Riemann moduli space. It proves ergodicity for the analogous lift of the Weil–Petersson geodesic local flow.

Flat bundles, von Neumann algebras andK-theory with ℝ/ℤ-coefficients

LetMbe a closed manifold andα:π1(M) →Una representation. We give a purelyK-theoretic description of the associated element in theK-theory group ofMwith ℝ/ℤ-coefficients ([α] ∈K1(M; ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relativeK-theory of the unital inclusion of ℂ into a finite von Neumann algebraB. We use the following fact: there is, associated withα, a finite von Neumann algebraBtogether with a flat bundleℰ→Mwith fibersB, such thatEα⊗ℰis canonically isomorphic with ℂn⊗ℰ, whereEαdenotes the flat bundle with fiber ℂnassociated withα. We also discuss the spectral flow and rho type description of the pairing of the class [α] with theK-homology class of an elliptic selfadjoint (pseudo)-differential operatorDof order 1.

Leafwise homotopy equivalences and leafwise Sobolov spaces

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L2 cohomology is an isomorphism which only depends on the leafwise homotopy class of the homotopy equivalence.

The Stokes Structure of a good meromorphic flat bundle

We give a survey on the Stokes structure of a good meromorphic flat bundle. We also show that a meromorphic flat bundle has the good formal structure if and only if it has a good lattice.