Random 3-manifolds and towers of their covers

Any closed 3-manifold can be obtained by glueing two handle bodies along their boundary. For a fixed such glueing, any other differs by changing the glueing map by an element in the mapping class group. Beginning with an idea of Dunfield and Thurston, we can use a random walk on the mapping class group to construct random 3-manifolds. I will report on recent work on the structure of such manifolds, in particular in view of tower of coverings and their topological growth: Torsion homology growth, the minimal degree of a cover with positive Betti number, expander families. I will in particularly explain the connection to some open questions about the mapping class group.