$L^2$-homology and Sigma invariants

Based on joint work with Dawid Kielak. In this talk I will give an overview of two families of invariants of a group: the Sigma invariants encode geometrically information about fibrings of $G$ over the integers $\mathbb Z$; $L^2$ homology on the other hand is an equivariant homology that accounts for all unitary representations of the group. A classical result of Wolfgang Lück shows that non-vanishing of $L^2$ homology obstructs the existence of mapping tori structures, here we will generalise this result to the BNSR invariants and discuss some of the difficulties that arise.