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

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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 Γ⊂ Isom X any homology class in H₁(Γ X,F₂) has a representative cycle of total length oₓ(Vol(Γ X)). As an application we show that ₂ H₁(Γ X,F₂)=oₓ(Vol(Γ X)).