Automorphism groups of discrete structures such as graphs carry a totally disconnected topology: both the groups and the structures they act on are thus zero-dimensional. The group topology is often non-discrete, and it has been found in recent years that there is a strong interplay between the algebraic and topological properties of these groups.
Totally disconnected locally compact groups are, it turns out, locally profinite and have a rich structure theory which has parallels with the theory of Lie groups, although it is more complicated.
This theory is still being developed and, in addition to Lie theory, draws on results about finite groups and from geometric group theory.
Summarising progress so far, an analysis of the locally profinite structure of the groups corresponds to the local theory of Lie groups, and a canonical form for group elements corresponds to eigendecomposition in the Lie algebra of a Lie group. A decomposition theory separates the cases of discrete and profinite groups, which are treated as negligible, from cases which are not negligible such as simple Lie groups over $p$-adic or function fields and automorphism groups of regular trees.
The techniques developed have been applied so far to answer questions about ergodic theory, random walks and arithmetic groups.