---
tag: HR5MB30MV3
kind: concept
name: Structure theory of totally disconnected locally compact groups
msc: [22D05, "20E08"]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_compact_group
---

## Summary

Every [[locally compact group|AHV05ZBCYW]] $G$ has a closed connected normal subgroup, its [[identity component|SDQTNBDJ2H]] $G^\circ$.
The quotient $G/G^\circ$ is a [[totally disconnected|AW3J9AQSA0]] 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|4E4RXKV9W3]] on compact open subgroups and the [[Cayley–Abels graphs|QWG66JE1JE]] of compactly generated groups.
Another is the scale function of an automorphism, introduced by George Willis.

## Explanations

### L1 human

Totally disconnected locally compact groups include all discrete groups, the $p$-adic groups such as $\mathrm{GL}_n(\Q_p)$, and the automorphism groups of locally finite trees.
Such a group has many small compact open subgroups, which play the role that small neighbourhoods play in Lie groups.
A compactly generated group of this kind acts nicely on a connected graph of finite degree, so it can be studied like a finitely generated group through its geometry.
The structure theory asks how such groups are built from simpler pieces, such as simple groups and compact groups.