Growth of mod$-2$ homology in higher rank locally symmetric spaces

Abstract

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 $\dim_{\mathbb F_2} H_1(\Gamma\backslash X,\mathbb F_2)=o_X({\rm Vol}(\Gamma\backslash X)).$