Let $X$ be a building of arbitrary type. A compactification ${𝒞}_{\mathrm{sph}}\left(X\right)$ of the set ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ of spherical residues of $X$ is introduced. We prove that it coincides with the horofunction compactification of ${\text{Res}}_{\mathrm{sph}}\left(X\right)$ endowed with a natural combinatorial distance which we call the root-distance. Points of ${𝒞}_{\mathrm{sph}}\left(X\right)$ admit amenable stabilisers in $\text{Aut}\left(X\right)$ and conversely, any amenable subgroup virtually fixes a point in ${𝒞}_{\mathrm{sph}}\left(X\right)$. In addition, it is shown that, provided $\text{Aut}\left(X\right)$ is transitive enough, this compactification also coincides with the group-theoretic...

Let $K$ be a $p$-adic field, and let $H=PS{L}_2\left(K\right)$ endowed with the Haar measure determined by giving a maximal compact subgroup measure $1$. Let $A{L}_H\left(x\right)$ denote the number of conjugacy classes of arithmetic lattices in $H$ with co-volume bounded by $x$. We show that under the assumption that $K$ does not contain the element $\zeta +{\zeta }^{-1}$, where $\zeta$ denotes the $p$-th root of unity over ${\mathbb{Q}}_p$, we have $$\underset{x\to \infty }{lim}\frac{logA{L}_H\left(x\right)}{xlogx}=q-1$$ where $q$ denotes the order of the residue field of $K$.