We prove a vanishing theorem in uniformly finite homology for the  genus of a complete spin manifold of bounded geometry and non-negative scalar curvature. This theorem is then applied to obstruct the existence of such metrics for some infinite connected sums, giving a converse to a theorem of Block and Weinberger.

Let λ_q be the irreducible representation of SL(2, ℝ) in SL(q, ℝ). Define a Fuchsian subgroup of SL(q, ℝ) to be a subgroup conjugate to a discrete subgroup of λ_q(SL(2, ℝ)). We prove in this paper that the fundamental group of a compact surface does not act properly on the affine space by affine tranformations if its linear part is Fuchsian.

Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations, is defined from "families of families" of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over ℙ¹. The complete list of geometric solutions to the Painlevé VI equation arising in this way is determined. Motivated by this construction, we define another class of algebraic isomonodromic deformations whose monodromy preserving families arise by "pullback" from (rigid) local systems. Using explicit methods from the theory of Hurwitz spaces, all such algebraic Painlevé VI solutions coming from arithmetic triangle groups are classified.

Let M be a compact, connected, orientable, hyperbolic 3-manifold whose boundary is a torus. We show that there are at most five slopes on $\partial M$ whose associated Dehn fillings have either a finite or an infinite cyclic fundamental group. Furthermore, the distance between two slopes yielding such manifolds is no more than three, and there is at most one pair of slopes which realize the distance three. Each of these bounds is realized when M is taken to be the exterior of the figure-8 sister knot.