The distortion dimension of Q -rank 1 lattices

Let X= G/ K be a symmetric space of noncompact type and rank k≥ 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 Q-rank-1 lattice Γ in a linear, semisimple Lie group G of R-rank k is k- 1. That is, given m< k- 1 , a Lipschitz m-sphere S in (a polyhedral complex quasi-isometric to) Γ , and a (m+ 1) -ball B in X (or G) filling S, there is a (m+ 1) -ball B^′ in Γ filling S such that vol B^′∼ vol B. In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension k- 1.