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 $\Gamma \subset \mathrm{Isom}X$ any homology class in $H_1(\Gamma \backslash X,{\mathbb{F}}_2)$ has a representative cycle of total length $o_X(\mathrm{Vol}(\Gamma \backslash X))$. As an application we show that ${\dim }_{{\mathbb{F}}_2}{H}_1(\Gamma \backslash X,{\mathbb{F}}_2)=o_X(\mathrm{Vol}(\Gamma \backslash X)).$

12 pages, 2 figures