Abstract Consider a stationary Poisson process in d -dimensional hyperbolic space. For $$R>0$$ R > 0 define the point $$\xi _R^{(k)}$$ ξ ( k ) of exce...

We present a proof of Poincaré’s Theorem on the existence of a Fuchsian group for any signature (g; m1, ..., mr), where g, m1, ..., mr provide an admissable solution to the formula describing the hyperbolic area of the group’s quotient space. Along the way we elucidate relevant concepts in hyperbolic geometry and the theory of Fuchsian groups.

We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that volM ≤ 3.44, then dimZ2 H1(M ;Z2) ≤ 7.

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.

Contents 1. Hyperbolic geometry 1 2. Fuchsian groups and hyperbolic surfaces 4 3. Spectrum and resolvent 11 4. Spectral theory: finite-area case 16 5. Spectral theory: infinite-area case 20 6. Selberg trace formula 23 7. Arithmetic surfaces 33 References 44 1. Hyperbolic geometry In complex analysis, we learn that the upper half-plane H = {Im z > 0} has a large group of conformal automorphisms,...

In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball [B(H)]−1 := n (X1, . . . ,Xn) ∈ B(H) n : ‖X1X ∗ 1 + · · ·+XnX ∗ n‖ 1/2 ≤ 1 o , where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and its implications to noncommutative function theory. The central object is an intertwining operator LB,A of the minimal isometric...

In this work, we study the asymptotic geometry of the mapping class group and Teichmuller space. We introduce tools for analyzing the geometry of “projection” maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the ...

Archimedes computed the center of mass of several regions and bodies [Dijksterhuis], and this fundamental physical notion may very well be due to him. He based his investigations of this concept on the notion of moment as it is used in his Law of the Lever. A hyperbolic version of this law was formulated in the nineteenth century leading to the notion of a hyperbolic center of mass of two point...

Given a lattice Veech group in the mapping class of closed surface S $S$ , this paper investigates geometry Γ $\Gamma$ associated π 1 $\pi _1S$ -extension group. We prove that is fundamental bundle with singular Euclidean-by-hyperbolic geometry. Our main result collapsing “obvious” product regions universal cover produces an action on hyperbolic space, retaining most . This key ingredient seque...