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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.

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    $\begingroup$ Projective representations are pretty useful... $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 22, 2011 at 18:25
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    $\begingroup$ Making a group act freely (or almost freely in some sense) on some contractible topological space is a standard way of studying the group. I am thinking especially of torsion-free groups, where the space has a chance of being a manifold or at least finite-dimensional. Some keywords: geometric group theory, classifying space, Teichmueller theory. $\endgroup$
    – Tom Goodwillie
    Commented Feb 22, 2011 at 18:53
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    $\begingroup$ The point of looking for actions on vector spaces is that vector spaces are particularly well-behaved, and so you can study a group by studying its linear representations. But groups most naturally act just on 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) $\endgroup$
    – Theo Johnson-Freyd
    Commented Feb 22, 2011 at 18:57
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    $\begingroup$ (continuation) On the other hand, its action through PSO(n) on the metrized real projective space (sphere mod antipodal identification) does not extend well to a linear representation, and gives a nice example of Mariano's comment that projective representations are cool. $\endgroup$
    – Theo Johnson-Freyd
    Commented Feb 22, 2011 at 18:59
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    $\begingroup$ @Theo: Actually, the action of PSO(n) on real projective space does extend quite naturally to a linear action: If you let SO(n) act by conjugation on the vector space of symmetric $n$-by-$n$ matrices, then $\mathbb{RP}^{n-1}$ is identified with the set of symmetric matrices~$p$ that satisfy $p^2=p$ and have trace equal to $1$. One identifies $p$ with its $+1$ eigenspace, which is a line in $\mathbb{R}^n$. $\endgroup$
    – Robert Bryant
    Commented Dec 13, 2011 at 15:55

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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.

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    $\begingroup$ On the other hand, one of the key tools for understanding / classifying the action of $G$ on a compact Riemann surface $X$ is via its associated representation on the space $H^0(X,\Omega^1)$ of global holomorphic differentials. So the representation theory is there lurking just beneath the surface. (Which is a good thing, IMO...) $\endgroup$
    – Pete L. Clark
    Commented Feb 23, 2011 at 14:06
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    $\begingroup$ A related example comes from the theory of K3 surfaces. Here, their automorphism groups can be infinite and are not "linearisable", by which I mean do not extend to an automorphism of projective space. However, one can understand this group action by studying its representation on the 2nd cohomology group. $\endgroup$
    – Daniel Loughran
    Commented Feb 23, 2011 at 14:32
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One of the most important way of getting Lie transformation groups (the one that motivated Sophus Lie in the first place) is to look at the group of symmetries of a smooth manifold with some "extra structure". For example, the group $G= $Isom$(M)$ of isometries of a complete Riemannian manifold $M$ is always a Lie group, and the associated Lie algebra consists of the Killing vector fields on $M$ with the usual bracket for vector fields. If $M$ is compact, then $G$ is also compact. A lot of the deeper theory of Lie groups and their homogeneous spaces $G/K$ come from this class of examples. Perhaps the most beautiful class are the so-called symmetric spaces of Cartan. These are the complete Riemannian manifolds $M$ such that at each point $p$ there exists an isometry that fixes $p$ and reverses all the geodesics through $p$. There are loads of great books on this subject, for example Helgason's ``Differential Geometry, Lie groups, and Symmetric Spaces".

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  • $\begingroup$ What that his motivation? I would hav imagined that looking at those groups was a way of constructing examples, and that he was more interested in symmetries of PDEs and other such things (which have much less structure, in a way...) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 22, 2011 at 18:58
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    $\begingroup$ @Mariano: In fact, Lie studied germs of analytic groups acting on germs of analytic manifolds---the concept of a global manifold and global group manifold did not exist in the middle and late 1800s when he did his pioneering work. And yes, you are correct that the "extra structure" that Lie dealt in his ``Geometrie Der Ber\"uhrungs Transformationen'' with was primarily differential equations. $\endgroup$
    – Dick Palais
    Commented Feb 22, 2011 at 19:07
  • $\begingroup$ Good point. Although symmetries of a (pseudo)Riemannian manifold are a daily bread of a theoretical physicist :) But I actually never studied homogeneous spaces and I guess I should fix that. $\endgroup$
    – Marek
    Commented Feb 22, 2011 at 19:28
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Bass-Serre theory studies groups via their action on trees. This is a combinatorial version of groups acting on simply connected spaces and leads to a very nice theory; one treats the quotient as a kind of orbifold and deduces information about the group from its structure.

Bruhat-Tits trees are a natural example of trees equipped with group actions, but I can't say much more about this.

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    $\begingroup$ 'This is a combinatorial version of groups acting on simply connected spaces...'. I'm a little mystified by this. It's a special case of a group acting on a simply connected space! See Scott and Wall's classic article 'Topological methods in group theory' for a geometric point of view on Bass--Serre theory. $\endgroup$
    – HJRW
    Commented Feb 22, 2011 at 21:30
  • $\begingroup$ Why are you mystified, HW? It is hardly news that special cases are often of great value! $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 13, 2011 at 15:47
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Finite group actions on sets have important applications to combinatorics, e.g., the Polya enumeration theorem.

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  • $\begingroup$ Well, I only had continuous groups in mind but this is actually very neat. $\endgroup$
    – Marek
    Commented Feb 22, 2011 at 19:31
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Zimmer's program is about continuous (or differentiable) actions of groups on manifold. Roughly, it expects that a lattice in a rank $r$ semi-simple Lie group cannot act non-trivially on a manifold of dimension $<r$. This result is known for the circle (see "Actions de réseaux sur le cercle" by Étienne Ghys, Inventiones 99) but, up to my knowledge, is still open even for surfaces.

More generally, many geometric, topological and dynamical problems are about group actions.

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In number theory there are countless examples. Off the top of my head, here are three:

  1. The group SL$_2({\mathbb Z})$ acts on the space of binary quadratic forms $ax^2 + bxy + cy^2$. The set of equivalence classes can be shown to form a group called the class group. By the way, the same group SL$_2$, even with coefficients from the reals, acts on the solution space of the Riccati equation $y' = a(x) + b(x)y + c(x)y^2$ and can be used to reduce the equation to some kind of "normal form".

  2. The group of rational points on an elliptic curve $E$ acts on certain curves of genus $1$ and makes them into principal homogeneous spaces. These are a most important tool for studying the group of rational points on $E$.

  3. The units of a quadratic number field act on the generators of principal ideals. In more archaic terms: solutions of the Pell equation $x^2 - dy^2 = 1$ act on the representations of a number $n$ as $n = x^2 - dy^2$. These kind of investigations lead to Dirichlet's class number formula, which is related to example 1.

  4. Galois groups tend to act on almost everything, but on the other hand have a habit of leading to representations and so belong to the list of examples you're less interested in.

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    $\begingroup$ For a number theorist... you count very badly! :D $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Dec 13, 2011 at 15:46
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Another important example is given by groups acting on graphs, especially (but not only) in finite group theory. Quite a few of the sporadic finite simple groups have actually been discovered as automorphism groups of graphs, e.g. the Hall-Janko group $J_2$ or the Higman-Sims group $HS$.

A related class of examples with more "structure" are so-called incidence geometries (combinatorial objects with geometric structure), and the most prominent example of those are the (Tits) buildings, introduced by Jacques Tits in the early 70's. (In fact, the example of Bruhat-Tits trees mentioned by Qiaochu Yuan is a very specific example of this situation; these are buildings of type $\tilde A_1$.)

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Groups can act on categories in ways that may be relevant to some physicists.

  1. One may consider a group acting on the derived category of coherent sheaves (also called the category of B-branes) of a complex manifold by exact autoequivalences. I think that if the manifold is an elliptic curve, the exact automorphism group contains the braid group $B_3$, which is substantially larger than the group of geometric automorphisms - there is an explanation in Polishchuk's book on Abelian varieties. I guess on the A side you can look for $A_\infty$-equivalences of Fukaya categories, but I don't know anything about that.

  2. In the geometric local Langlands program, a loop group $G((t))$ acts on categories of $\mathfrak{g}((t))$-modules attached to opers, where $G$ is a linear algebraic group. (An oper is a kind of $G$-connection on a curve with some extra structure - see E. Frenkel's book Langlands for loop groups).

  3. More concretely, if a group acts by automorphisms on an algebra $A$ over the complex numbers, then it also acts on the category of $A$-modules. By Schur's lemma, an irreducible $A$-module then inherits an action of a central extension of its objectwise stabilizer.

  4. A manifestation of the previous example that is close to my heart is the case when the monster simple group acts by automorphisms on the monster vertex algebra (which isn't quite an algebra, but the same idea applies), and hence on the categories of twisted modules. We naturally get projective actions of large finite groups on irreducible twisted modules.

One has to be a little careful about what one means by an action of a group on a category, since there is the question of whether the composition of functors $F_g \circ F_h$ is equal to $F_{gh}$ or just naturally isomorphic, and whether associativity holds on the nose or up to some other system of isomorphisms that satisfies a pentagon identity. The things that naturally act on categories are called 2-groups, and groups that we see acting are a sort of "shadow" or truncation of them.

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  • $\begingroup$ Minor correction: The categories in number 2 are made out of projective $\mathfrak{g}((t))$-modules at the critical level. By "critical level", one means the central line of the central extension acts in a way that gives a completion of the enveloping algebra a very large center (isomorphic to the coordinate ring of a space of opers). $\endgroup$
    – S. Carnahan ♦
    Commented Feb 23, 2011 at 16:46
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In addition to what has been said already: I think that everywhere in Mathematics when you speak of symmetries you mean "group plus action" and not just the group itself.

The thoughts about symmetry are probably of geometric nature: asking for symmetry means asking for the symmetry of a geometric object (we have already the examples of Riemannian manifolds, but there are many more) In differential geometry you can ask for "symmetries" of all kind of structures: metric, but also symplectic forms or Poisson tensors. In this case you enter the realm of dynamical systems with symmetries. The symmetries usually help to simplify the dynamical system by using "conserved quantities" to eliminate degrees of freedom. You may remember this from your first mechanics courses when dealing with the Kepler problem...

But symmetries in crystals might yet give another example, not related to Lie groups and some inherited action from a linear action: treating a crystal as an abstract lattice with colored edges and vertices one may well ask for its symmetries and arrives at discrete groups acting in a much more combinatorial way. The original possibility that the lattice can be embedded into some Euclidean space is no longer relevant.

In addition, symmetries arise in much more abstract concepts that these geometric ones. A prominent example is perhaps the question of solving polynomial equations. Here the symmetries of the polynomial might allow for general formulas or not. This is the beginning of Galois theory in field theory, where not Lie groups but discrete groups are acting.

From my own field a statement which I would like to understand better: the Grothendieck-Teichmueller group acts on the set of Drinfeld associators. Not a linear action at all :(

On the other hand: One reason why linear actions are so omnipresent is perhaps that (beside being the simplest type of actions) all types of geometric actions dualize to a linear action on the spaces of reasonable functions on the geometric spaces. Hence even a group action on some geometric object (manifold, lattice, ...) can by studied by means of representation theory when one looks for the induced action (via pull-back) on the functions on it. However, this is typically quite complicated as the representation spaces typically are infinite-dimensional.

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  • $\begingroup$ I am quite familiar with all the stuff you've said (except for those Drinfeld associators, those left me baffled) but it's good to be reminded of it. Also, the last remark is particularly valid, I think. I am only acquainted with Peter-Weyl theorem for compact groups but I appreciate the idea that one can get a lot just by looking at the regular representation, or group algebra or some similar canonically associated object. $\endgroup$
    – Marek
    Commented Feb 22, 2011 at 20:07
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There are interesting actions, called affine isometric actions, that are related to orthogonal representations (roughly they are "perturbations" of orthogonal representations by certain cocycles into the representation space), and arise in an essential way in the study of Kazhdan's property (T).

This is treated in depth in the following fantastic book on Property (T):

http://perso.univ-rennes1.fr/bachir.bekka/KazhdanTotal.pdf

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