On the kissing number of hyperbolic surfaces with a small systole

Parlier proved that there exists a constant $U>0$ such that for any hyperbolic surface $S$ of genus $g$ with systole $\mathrm{\ell }$ the following holds: $$Kiss(S)\le U\frac{e^{\mathrm{\ell }}}{\mathrm{\ell }}g$$ where $Kiss(S)$ is the maximum number of homotopically distinct systoles in $S$ (Kissing number). We show that there exists a constant $V>0$ and a sequence $S_n$ of hyperbolic homogeneous surfaces of genus $g_n\to \mathrm{\infty }$ with a bounded systole such that $$Kiss(S_n)\ge V{g}_n$$ for all $n\ge 1$. This shows that Parlier’s theorem is sharp for a fixed systole length. Our proof uses Bruhat–Tits buildings of type ${\tilde{A}}_2$ and Hamiltonian surfaces.