Interesting topological groups that aren't manifolds?

Question

Does anyone know of examples of "pathological" topological groups that are actually used? I am particularly interested in infinite non-Hausdorff examples.

Answer 1

A large and common class of topological groups that aren't Lie groups are profinite groups. These naturally appear in Galois theory as Galois groups of infinite extensions, in number theory in relation to $p$-adic Lie groups, and in algebraic geometry as étale fundamental groups.