Structure theory of totally disconnected locally compact groups

Summary

Every locally compact group $G$ has a closed connected normal subgroup, its identity component $G^\circ$. The quotient $G/G^\circ$ is a totally disconnected locally compact group. Connected locally compact groups are close to Lie groups by the solution of Hilbert's fifth problem. So the study of general locally compact groups leads to the structure theory of totally disconnected ones. Its basic tools are van Dantzig's theorem on compact open subgroups and the Cayley–Abels graphs of compactly generated groups. Another is the scale function of an automorphism, introduced by George Willis.