In my doctoral dissertation (directed by W. P. Thurston) I studied the geometry of convex polyhedra in hyperbolic 3-space H3, and succeeded in producing a geometric characterization of dihedral angles of compact convex polyhedra by reducing the question to a convex isometric embedding problem in the De Sitter sphere, and resolving this problem. In particular, this produced a simple alternative ...

In [Comput. Math. Appl. 41 (2001), 135–147], A.A. Ungar employs the Möbius gyrovector spaces for the introduction of the hyperbolic trigonometry. This Ungar’s work plays a major role in translating some theorems from Euclidean geometry to corresponding theorems in hyperbolic geometry. In this paper we explore the theorems of Stewart and Steiner in the Poincaré disc model of hyperbolic geometry.

This work brings to light some anabelian behaviours of analytic curves in the context Berkovich geometry. We show that knowledge tempered fundamental group called analytically determines their skeletons as graphs. The famous Drinfeld half-plane is an example such a curve. space, introduced by André, enabled Mochizuki prove first result geometry, dealing with analytifications hyperbolic over ℚ ¯...

We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart of this statement is that in the case of unbounded combinatorics renormalized mappings become almost quadratic. Technically, this follows from the decay of the box geometry. Specific estimates of the rate of this decay are shown which are sharp in a class of...

We prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in R and C and we discuss the situation of curves in the Riemann sphere C ∪ {∞}. ...