Packings by translation balls in $${\widetilde{{\rm SL}_2({\mathbb{R}})}}$$ SL 2 ( R ) ~

Abstract

For one of Thurston model spaces, \({\widetilde{{\rm SL}_2({\mathbb{R}})}}\), we discuss translation balls and packing that space by such balls in contrast to the packing by standard (geodesic) balls. We present an infinite family of packings generated by discrete groups of isometries, and observe numerical results on their densities. In particular, we found packings whose densities are close to the upper bound density for ball packings in the hyperbolic 3-space.