Minimal Submanifolds and Waists of Locally Symmetric Spaces

We show that compact locally symmetric manifolds $M$ with universal cover the symmetric space $X$ for $SL(n,\mathbb{R})$ form a topological higher $d$-expander family for $d\leq n/8$. We prove the same statement for $SL(n,\mathbb{R})$ replaced by a split simple non-compact real Lie group $G$ and for