Geodesic and Translation Ball Packings Generated by Prismatic Tessellations of the Universal Cover of $${{\rm {SL}}_{2}({\mathbb{R}})}$$ SL 2 ( R )

Abstract

We construct ball packings of the universal cover of $${{\rm{SL}}_{2}({\mathbb{R}})}$$ by geodesic balls and translation balls. The packings are generated by action of the prism groups $${{\mathbf{pq}}_{k}{\mathbf{o}}_{\ell}}$$ . We obtain volume formulae for calculations in geographical coordinates. Using these formulae we find numerically the maximal dense packings for cases $${k=1}$$ , $${o=2}$$ , $${\ell=1}$$ and small values of $${p}$$ and $${q}$$ .