On which structures does the free group 'naturally' act?

One of the best ways to get a handle on a group is to recognize it as isomorphic to a set of symmetries of some structure. The dihedral group of order $2n$ is easily recognized as the set of symmetries of a regular $n$-gon, the symmetric group as the permutations of a set, the Klein-four group as the symmetries of the ‘cross’—one pair of opposing prongs are longer than the other pair.

The previous descriptions are also in a sense ‘exhaustive’: every ‘sensible’ symmetry you could conceive of those structures is represented by some element of the group. An example where the group is not exhaustive over its structure is $C_n$ over the regular $n$-gon (reflection is missing), or $A_n$ over the set of $n$ elements.

I wish to know of a structure that a free group—specifically $F_2$—naturally acts on which it is also in a sense ‘exhaustive’. I’m aware of $F_2$’s Cayley graph, but I want other examples.


The question may have been a bit unfair and imprecise, but I can’t help that it is really.
One of the reasons it may be unfair is that ‘exhaustive’ is a bit subjective. For example, I myself cannot point out a structure that $C_n$ naturally acts on that $D_{2n}$ could not be conceived of as acting on also; however, both of these groups also sit in $S_n$ in a more-or-less canonical fashion. But there are structures, such as polygons, that we think of as $D_{2n}$ acting on naturally rather than $S_n$.

I’m trying to find structures such that when we think of their ‘natural’ set of symmetries, the group of these symmetries would be isomorphic to a free group. I want a
couple of examples, as I’m slightly disappointed in having only the Cayley graph or finite tuples of words as a reference. Other groups have many examples of acting on a structure. For example the Klein four group could also be realized as the symmetries of a proper rectangle, $A_4$ can be realized as the rigid motions of the tetrahedron rather than the even permutations of a 4-element set.

I also don’t want these examples to be ‘cheap’. What I mean by this is that the action is not arrived at by quotienting by a normal subgroup to get an action that is more properly realized by another group.

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$F_2$ acts on a certain tiling of the hyperbolic plane. It looks sort of like this:


The above tiling is acted on by the modular group $\Gamma \cong \text{PSL}_2(\mathbb{Z})$, which naturally sits as a subgroup inside of the full group $\text{PSL}_2(\mathbb{R})$ of isometries of the hyperbolic plane. Abstractly, this group is the free product $C_2 \ast C_3$. It has congruence subgroups $\Gamma(N)$ given by the image of the kernel of the quotient maps $\text{SL}_2(\mathbb{Z}) \to \text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ in $\Gamma$. For $N \ge 2$ these groups act freely on the hyperbolic plane, and the topological structure of their quotients are known (they are uncompactified modular curves, all of which will be compact Riemann surfaces minus finitely many points, and both the genus and the number of points removed are known). In particular, $\mathbb{H}/\Gamma(2)$ is known to be $\mathbb{R}^2$ minus two points, which has fundamental group $F_2$. Hence

$$\Gamma(2) \cong F_2$$

acts on the hyperbolic plane $\mathbb{H}$. You can find a fundamental domain for this action by taking a union of finitely many fundamental domains for the action of $\Gamma$ (pictured above), and this gives a tiling of the hyperbolic plane whose automorphism group should be exactly $\Gamma(2)$ (I think).

Groups that arise in this way can be thought of as hyperbolic analogues of wallpaper groups.

In general, an interesting way to find an interesting space on which a group $G$ acts is to find a space $X$ with $\pi_1(X) \cong G$ and look at the action of $G$ on the universal cover $\tilde{X}$, especially if $X$ has extra structure for its universal cover to inherit. In the case of $F_2$ we can pick $X$ to be the wedge of two circles and this reproduces the action of $F_2$ on its Cayley graph, but for other kinds of spaces $X$ we get more interesting spaces. Above we instead pick $X$ to be $\mathbb{R}^2$ minus two points and get an action on the hyperbolic plane; this is a special case of uniformization.

Edit: Perhaps an easier description of what I’m talking about is in terms of the von Dyck groups. In the Poincar√© disk model the relevant tiling looks like this:

enter image description here

$F_2$ is (Edit:) the group of orientation-preserving symmetries of this tiling that can be obtained by rotating around the vertices of any of the triangles in it. (The original statement that was here was wrong; the tiling above clearly admits a rotational symmetry of order $3$, but $F_2$ has no torsion. The resolution is that this rotational symmetry fixes the origin, which is not a vertex.)