Hyperbolic structures are obtained by tiling a hyperbolic surface with negative Gaussian curvature. These structures generally exhibit two percolation transitions: a system-wide connection can be established at a certain occupation probability p = pc1, and there emerges a unique giant cluster at pc2 > pc1. There have been debates about locating the upper transition point of a prototypical hyper...

The current work focuses on critical exponents of the two-dimensional ferromagnetic Ising model defined on the hyperbolic plane. The hyperbolic plane is a simply connected infinite surface in which the Gaussian curvature possesses a constant negative curvature at arbitrary points. This non-zero curvature changes the geometric symmetry of the embedded Ising lattice model, and thus possibly induc...

Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation Mτ in a 2 + 1 dimensional flat spacetime V with compact hyperbolic Cauchy surfaces satisfies limτ→−∞ lMτ = sT , where lMτ and sT denote the marked length spectrum ofMτ and the marked measure spectrum of the R-tree T , dual to the measured foliation corresponding to the translational part of the holonomy of ...

Let V be a maximal globally hyperbolic flat n+1–dimensional space–time with compact Cauchy surface of hyperbolic type. We prove that V is globally foliated by constant mean curvature hypersurfaces Mτ , with mean curvature τ taking all values in (−∞, 0). For n ≥ 3, define the rescaled volume of Mτ by H = |τ | Vol(M, g), where g is the induced metric. Then H ≥ nVol(M, g0) where g0 is the hyperbol...

Let Γ be a torsion-free lattice in SO0(3, 1), and let M = Γ\H3 be the corresponding hyperbolic 3-manifold. It is wellknown that in the presence of a closed, embedded, totallygeodesic surface in M , the canonical flat conformal structure on M can be deformed via the bending construction. Equivalently, the lattice Γ admits non-trivial deformations into SO0(4, 1). We present a new construction of ...

Let ν+ and ν− be two measured laminations which fill up a hyperbolic surface. Kerckhoff [Duke Math. J. 65 (1992), 187–213] defines a line of minima as a family of surfaces where convex combinations of the hyperbolic length functions of ν+ and ν− are minimum. This is a proper curve in the Teichmüller space. We show that there exists a line of minima which does not converge in the Thurston compac...

Let M be a (connected) hyperbolic 3–manifold, namely a complete Riemannian manifold of dimension 3 and of constant sectional curvature −1, with finitely generated fundamental group. A fundamental subset of M is its convex core CM , which is the smallest non-empty convex subset of M . The condition that the volume of CM is finite is open in the space of hyperbolic metrics on M , provided we rest...

Our main theorem is that, if M is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre S and pseudo-Anosov monodromy, then the lift of the inclusion of S in M to universal covers extends to a continuous map of B to B , where B D H [ S 1 1 . The restriction to S 1 maps onto S 1 and gives an example of an equivariant S –filling Peano curve. After proving the main theo...

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon’s Conjecture: A hyperbolic group G (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic ...

We show that for each aspherical compact complex surface X whose fundamental group π fits into a short exact sequence 1 → K → π → π1(S) → 1 where S is a compact hyperbolic Riemann surface and the group K is finitelypresentable, there is a complex structure on S and a nonsingular holomorphic fibration f : X → S which induces the above short exact sequence. In particular, the fundamental groups o...