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.
Best Answer
Finite group actions on compact Riemann surfaces are a classical subject, and the related literature is huge.
It is well known that if a finite group $G$ acts as a group of automorphisms on a compact Riemann surface of genus $g \geq 2$, then necessarily
$|G| \leq 84(g-1)$.
This is a old result of Hurwitz, and if equality holds then the group $G$ is called a Hurwitz group in genus $g$. The classification of Hurwitz groups is not yet completed; it is known that there exists a Hurwitz group for infinitely many values of $g$, and that there exists no Hurwitz group for infinitely many values of $g$ as well.
Moreover, any Hurwitz group $G$ is a quotient of the infinite triangle group
$T_{2,3,7}=\langle x, y | x^2=y^3=(xy)^7=1 \rangle$.
There exist no Hurwitz group in genus $2$, and exactly one in genus $3$. It is the group $G=PSL(2, \mathbb{F}_7)$, the unique simple group of order $168$. The corrisponding Riemann surface can be realized as a particular curve of degree $4$ in $\mathbb{P}^3(\mathbb{C})$, the so-called Klein quartic.