In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3–manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed, orientable, hyperbolic 3–manifolds that have the same set of surfaces.

We analyzed surface-wave propagation that takes place at the boundary between a semi-infinite dielectric and a multilayered metamaterial, the latter with indefinite permittivity and cut normally to the layers. Known hyperbolization of the dispersion curve is discussed within distinct spectral regimes, including the role of the surrounding material. Hybridization of surface waves enable tighter ...

In this paper we show that there is an upper bound on the volume of a hyperbolic knot in the 3-sphere with canonical genus g. This bound can in fact be chosen to be linear in g. In other words, if Seifert's algorithm builds a surface with small genus for a hyperbolic knot, then the complement of the knot cannot have large

In a talk at the 1954 International Congress,2 the author outlined an existence proof for a surface of the form z=f(x, y) which covers every point of the x, y-plane exactly once, and which is conformally equivalent to the interior of the unit circle.3 This paper gives a simple method for the explicit construction of such a surface. It is based on the following observations regarding the type of...

We construct first examples of discrete geometrically finite subgroups of PU(2, 1) which contain parabolic elements, and are isomorphic to surface groups of genus ≥ 2.

In this paper a hyperbolic Riemann surface is described, and the uniformizing function for this surface is constructed. The uniformizing function for this hyperbolic surface is an infinite product similar to the ones constructed for some parabolic surfaces in earlier studies [1], [2], [4]. Description of the surface. Let I denote the positive integers. For each iEI, let C, be the vertical line ...

Herein we study propagation of surface waves at a boundary of an isotropic media and a multilayered hyperbolic metamaterial. The structure dispersion is discussed for various cases of a hyperbolic metamaterial. It is demonstrated that it is possible to tune the frequency range of surface waves by varying the thickness of dielectric sheets. It is also shown that this frequency range can be broad...

A subgroup of a group is known as a surface subgroup if it is isomorphic to the fundamental group of a closed connected orientable surface with positive genus. The surface subgroup conjecture in low-dimensional topology proposes that if the fundamental group of a closed orientable irreducible 3-manifold is infinite, then it contains a surface subgroup. With the recent solution to the geometrisa...

Let M by a complete hyperbolic 3-manifold homotopy equivalent to a compact surface Σ. Let Φ be a proper subsurface of Σ, whose boundary is sufficiently short in M . We show that the union of all Margulis tubes and cusps homotopic into Φ lifts to a uniformly quasiconvex subset of hyperbolic 3-space. 2010 Mathematics Subject Classification: 57M50

We study certain groups of isometries of hyperbolic space (including certain hyperbolic Coxeter groups) and normal subgroups with quotient Zν , ν ≥ 1. We show that the associated relative growth series are not rational. This extends results obtained by Grigorchuk and de la Harpe for free groups and Pollicott and the author for surface groups.