Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius manifold structures on Hurwitz spaces (real doubles, deformations) is described at the level of the corresponding Riemann-Hilbert problems

The concept of cohomological goodness was introduced by J.-P. Serre in his book on Galois cohomology [31]. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove, for example, that the Bianchi groups (i.e., the groups $\mathrm{PSL}(2,O)$, where $O$ is the ring of integers in an imaginary quadratic number field) are good. As an application of our improved understanding of goodness, we are able to show that certain natural central extensions of Fuchsian groups are residually finite. This result contrasts with examples of P. Deligne [5], who shows that the analogous central extensions of $\mathrm{Sp}(4,\mathbb{Z})$ do not have this property

Let $\Sigma$ be a compact surface of type $(g,n)$, $n>0$, obtained by removing $n$ disjoint disks from a closed surface of genus $g$. Assuming that $\chi (\Sigma )<0$, we show that on $\Sigma$, the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the $C^{\infty }$-topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak [OPS3, Theorem 2] for surfaces of type $(0,n)$ whose examples include bounded plane domains.

Our main ingredients are as follows. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on $\Sigma$. Second, we show that the space of such metrics is homeomorphic (in the $C^{\infty }$-topology) to the space of flat metrics (on $\Sigma$) with constantly curved boundary. Because of this, we next reduce the complicated degenerations of flat metrics to the simpler and well-known degenerations of hyperbolic metrics, and we show that determinants of Laplacians of flat metrics on $\Sigma$, with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because Khuri [Kh] showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when $\Sigma$ is of type $(g,n), g>0$, while Osgood, Phillips, and Sarnak [OPS3] showed the properness when $g=0$

We investigate the geometric propagation and diffraction of singularities of solutions to the wave equation on manifolds with edge singularities. This class of manifolds includes, and is modeled on, the product of a smooth manifold and a cone over a compact fiber. Our main results are a general diffractive theorem showing that the spreading of singularities at the edge only occurs along the fibers and a more refined geometric theorem showing that for appropriately regular (nonfocusing) solutions, the main singularities can only propagate along geometrically determined rays. Thus, for the fundamental solution with initial pole sufficiently close to the edge, we are able to show that the regularity of the diffracted front is greater than that of the incident wave