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

Abstract

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 $${{\mathrm{vol}}}B'\sim {{\mathrm{vol}}}B$$ . In particular, such arithmetic lattices satisfy Euclidean isoperimetric inequalities up to dimension $$k-1$$ .