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