The distortion dimension of $\mathbb Q$--rank $1$ lattices

Let $X=G/K$ be a symmetric space of noncompact type and rank $k\ge 2$. We prove that horospheres in $X$ are Lipschitz $(k-2)$--connected if their centers are not contained in a proper join factor of the spherical building of $X$ at infinity. As a consequence, the distortion dimension of an irreducible $\mathbb{Q}$--rank-$1$ lattice $\Gamma$ in a linear, semisimple Lie group $G$ of $\mathbb R$--rank $k$ is $k-1$. That is, given $m< k-1$, a Lipschitz $m$--sphere $S$ in (a polyhedral complex quasi-isometric to) $\Gamma$, and a $(m+1)$--ball $B$ in $X$ (or $G$) filling $S$, there is a $(m+1)$--ball $B'$ in $\Gamma$ filling $S$ such that $\operatorname{vol} B'\sim \operatorname{vol} B$. In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension $k-1$.