In this talk we are discussing finiteness/compactness properties for locally compact groups, extending what it means to be finitely ('compactly') generated and finitely ('compactly') presented. We start with a historical overview and then what is currently known about groups with specified compactness properties, focusing on the totally disconnected case.
We shall then learn a new construction of Artin-like tdlc groups. These allow us to obtain: a new family of (discrete) Thompson-like groups containing, for every n, groups of homological type FP_n but not of type FP_{n+1}; and a family of tdlc RAAGs acting nicely on cubed complexes, which are then of type FP_infinity in the tdlc sense.
Based on ongoing joint work with Ilaria Castellano, Bianca Marchionna, and Brita Nucinkis.