We exhibit the first examples of hyperbolic three-manifolds for which the Seiberg-Witten equations do not admit any irreducible solution. Our approach relies on hyperbolic geometry in an essential way; it combines an explicit upper bound for the first eigenvalue on coexact 1-forms λ_1^* on rational homology spheres which admit irreducible solutions together with a version of the Selberg trace formula relating the spectrum of the Laplacian on coexact 1-forms with the volume and complex length spectrum of a hyperbolic three-manifold. Using these relationships, we also provide precise numerical bounds on λ_1^* for several hyperbolic rational homology spheres.

For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

In this article we define new flows on the Hitchin components for PGL(V). Special examples of these flows are associated to simple closed curves on the surface and give generalized twist flows. Other examples, so called eruption flows, are associated to pair of pants in S and capture new phenomena which are not present in the case when n=2. We determine a global coordinate system on the Hitchin component. Using the computation of the Goldman symplectic form on the Hitchin component, that is developed by two of the authors in a companion paper to this article (Sun and Zhang in The Goldman symplectic form on the PGL(V)-Hitchin component, 2017. arXiv:1709.03589), this gives a global Darboux coordinate system on the Hitchin component.

In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.