What one group action can tell about another [closed]

Question

There are two groups, $G_1$ and $G_2$. They are both acting on a set $S$.

$S$ may have some structure. The groups may too. The actions respect them.

$G_1$ is mysterious. Perhaps all we know about it is the way it acts on $S$. We'd like to know more.

$G_2$ is well-known.

We might be able to learn about $G_1$ from watching how its action interacts with the action of $G_2$. Intersection of their orbits, for instance.

Is this situation systematically studied under some name? Beyond the case when the actions commute.

Answer 1

Whether this counts as learning more about the groups themselves is questionable, but: if $G_1$ and $G_2$ are both finite $p$-groups and $S$ is finite then we have $|S^{G_1}| \equiv |S| \equiv |S^{G_2}| \bmod p$ by two applications of the "$p$-group fixed point theorem." This is famously used in a very short proof of Fermat's two-square theorem due to Zagier, the details of which you can find e.g. in this blog post, and which is also the subject of an old MO question.

Answer 2

The Schwarz--Milnor lemma seems like a good example of what you're asking for. (But, as per my comment, is only really useful in certain contexts, i.e. infinite discrete groups.)

It applies when your "set" $S$ is a proper, geodesic metric space, and your groups $G_i$ are acting properly discontinuously and cocompactly by isometries. (For instance, this will be true of the fundamental groups of two compact metric spaces with isometric universal covers.)

The conclusion is that $G_1$ and $G_2$ are quasi-isometric.

This can enable you to deduce quite a lot of information about $G_1$ from $G_2$. For instance, you can conclude that $G_1$ is finitely generated.

More generally, geometric group theorists have devoted a great deal of effort to proving quasi-isometric rigidity theorems, which then enable you to say something about $G_1$. Famous, and very deep, examples include Gromov's polynomial growth theorem, which handles the case of nilpotent groups, Stallings' ends theorem for free groups, and the convergence group theorem of Casson--Jungreis and Gabai, which handles the case of hyperbolic surface groups.

In the best possible cases, you get to conclude that $G_1$ and $G_2$ are commensurable, meaning that they have isomorphic subgroups of finite index.