As a person interested in group theory and all things related, I'd like to deepen my knowledge of group actions.

The typical (and indeed the most prominent) example of an action is that of a representation. In this case the target space has so much structure that one can deduce a huge number of properties of a given group just by working out some linear algebra (to put it bluntly).

Now, I am wondering

whether there exist other structures that provide interesting classes of actions. Either for the study of the given group or just as an application to solve some interesting problems.

I realize my question is probably a bit naive and ignorant of what is probably a standard knowledge but I actually can't think of that many useful actions and wikipedia article on them doesn't provide many examples (at least not very interesting and non-linear). Not that I can't think of anything at all. Coming from physics, I am aware of stuff such as gauge symmetries (free transitive fiber-wise actions on fiber bundles) or various flows (whether for time evolution, or as an symmetry orbit). And I am also aware of the usual Lie theory, left/right translations, etc. But I am looking for more.

*Note:* feel free to generalize the above to any action. I'd be certainly also interested in actions of algebras, rings, etc.

spaces, and Lie groups, being groups of manifolds, naturally act on manifolds. SO(n), for example, by construction acts on the (n-1)-dimensional metrized sphere. That you can extend this to an n-dimensional linear representation is almost an accident (but sure does make studying it easier!). (continued) $\endgroup$3more comments