How does the volume control the topology of locally symmetric spaces?

Locally symmetric spaces are immensely interesting objects that lie at the crossroads of number theory, representation theory and homogeneous dynamics. Since the work of DeGeorge and Wallach in '70s and Gromov in '80s it has been known that many topological invariants of locally symmetric spaces are controlled only in terms of volume. One of the most ambitious conjectures in that direction was proposed by Gelander. He conjectured that any arithmetic locally symmetric space M is homotopy equivalent to a simplicial complex where the degree of vertices are bounded in terms of dimension of M and the total number of cells is linear in the volume of M. I will present some of the recent results on the growth of topological invariants of locally symmetric spaces and explain the main ideas behind the proof of Gelander's conjecture for hyperbolic 3-manifolds.