Kissing number in hyperbolic space

Abstract

This paper provides upper and lower bounds on the kissing number of congruent radius $r>0$ spheres in ${\mathbb{H}}^n$, for $n\ge 2$. For that purpose, the kissing number is replaced by the kissing function $\kappa (n,r)$ which depends on the radius $r$. After we obtain some theoretical lower and upper bounds for $\kappa (n,r)$, we study their asymptotic behaviour and show, in particular, that ${\lim }_{r\to \infty }\frac{\log \kappa (n,r)}{r}=n-1$. Finally, we compare them with the numeric upper bounds obtained by solving a suitable semidefinite program.

Will be merged with arXiv:1910.02715