  Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of
dimension at least $2$ such that the isometry group of $X$ has property $(T)$.
We prove that for every torsion free lattice $\Gamma\subset {\rm Isom} X$ any
homology class in $H_1(\Gamma\backslash X,\mathbb F_2)$ has a representative
cycle of total length $o_X({\rm Vol}(\Gamma\backslash X))$. As an application
we show that $\dim_{\mathbb F_2} H_1(\Gamma\backslash X,\mathbb F_2)=o_X({\rm
Vol}(\Gamma\backslash X)).$
