A Poincaré ball model for taxicab hyperbolic geometry

Abstract

Taxicab space is a modification of Euclidean space that uses an alternative notion of distance. Similarly, the Poincaré ball is a model of hyperbolic geometry that consists of a subset of Euclidean space with an alternative notion of distance. In this paper, we merge these two variations to create a taxicab version of the Poincaré ball. We determine the isometry group for this new space and show that this space is hyperbolic in the sense of Gromov.

24 pages, 8 figures