Realizing groups as symmetry groups

Question

We're supposed to think of (non-Abelian) groups as groups of symmetries of some object. Sometimes it isn't obvious what this object is. For example, the fundamental group of a topological space acts by symmetries on the universal cover. Does anyone have any examples of (non-Abelian) groups which aren't defined as the group of symmetries of something but turn out to be the group of symmetries of some non-obvious thing?

Edit: In response to Qiaochu's comment I have edited the question appropriately.

Answer 1

This is an abelian example, but: the center $Z(G)$ of a group $G$ turns out to have a non-obvious interpretation as the group of symmetries of something. Namely, it's the group of natural automorphisms of the identity functor $\text{id}_G$, thinking of $G$ as a category with one object. More generally one can define the center of an object in a $2$-category and this reproduces a few interesting constructions, e.g. the second homotopy group $\pi_2$; see this blog post for details.

Answer 2

If $G$ is an interesting group with an interesting normal subgroup $H$, and $H$ isn't defined as the kernel of some action of $G$, then $G/H$ is an interesting group that may not obviously act on anything. For example, the outer automorphism group of a group $G$ has this property. It acts on the conjugacy classes of $G$ as well as the irreducible representations of $G$, but not necessarily faithfully in either case.

The mapping class group $\text{MCG}(M)$ of, say, a smooth manifold $M$ is also defined in this way (and see also the Dehn-Nielsen theorem). It doesn't act on $M$. It acts, not necessarily faithfully, on various invariants associated to $M$, e.g. its (co)homology or unbased homotopy classes of loops.

If $M$ is a compact orientable surface, the subgroup of the mapping class group that preserves orientation acts on the Teichmüller space of $M$. Teichmüller space is itself a complex manifold, and if $M$ has genus greater than $1$, then this subgroup is precisely the group of biholomorphic automorphisms of Teichmüller space.

Answer 3

First let me give some general examples that work for a large class of groups (say countable discrete groups.)

1.) Given a measure space say $([0, 1], \mu)$ the unit interval with Lebesgue measure and a group $\Gamma$, construct the product measure space $([0, 1]^\Gamma, \mu^\Gamma)$. Then $\Gamma$ acts on the product by shifting the indices. This is a fundamental example in ergodic theory.

2.) Similarly consider the infinite tensor product of matrices index by $\Gamma$, $\bigotimes_\Gamma M_2(\mathbb{C})$. Then as above $\Gamma$ can act by acting on the index set. This is a key example in operator algebras.

3.) Any such $\Gamma$ is a fundamental group of a topological space, called $K(\Gamma, 1)$.

Now more specifically:

4.) $SL_2(\mathbb{R})$ is defined as $2\times 2$ matrices over the reals with determinant 1. So by definition it acts on $\mathbb{R}^2$. The more interesting action, which is not obvious from the definition is that it acts by conformal isometries on the hyperbolic plane. (In fact this might be THE most studied action of a group in all of mathematics)

5.) The outer automorphism group of a free group $Out(\mathbb{F}_n)=Aut(\mathbb{F}_n)/Inn(\mathbb{F}_n)$ is defined as a quotient so it doesn't have an obvious action. However, it does act on a cell complex called Outer Space.