We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a " hyperbolic non-percolation " property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbol...

We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a “hyperbolic non-percolation” property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbolic...

This paper introduces several cellular automata in hyperbolic spaces which are able to simulate any computational device, as they are universal. We sketchily present five universal cellular automata, four of them on two grids of the hyperbolic plane and one in a grid of the hyperbolic 3D space. Key–Words: Theory of Computation, Cellular Automata, universality, hyperbolic geometry.

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. ...

We study the algebraic and facial structures of hyperbolic programs, and examine natural relaxations of hyperbolic programs, the relaxations themselves being hyperbolic programs.

1. If a closed irreducible 3-manifold has word hyperbolic fundamental group, does it admit a Riemannian metric of constant negative curvature? 2. Is every nitely presented subgroup of a word hyperbolic group word hyperbolic? (Noel Brady Brady] has spoken on an example of nitely presented subgroup H of a word hyperbolic group G where cd(G) = 3 and H is not hyperbolic. I had shown previously that...

We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space – the natural extension o...