Proof. By Remark 1.2, we can assume $\Gamma \curvearrowright X$ is isometric.

For $R>0$ , define the R-connected components of X to be the maximal sets which cannot be written as a disjoint union of non-empty open sets which are distance $\geq R$ from each other. Since each such component is at least distance R from its complement in X, each is a clopen set. Notice that belonging to the same R-connected component of X is an equivalence relation on X and that the action $\Gamma \curvearrowright X$ permutes these equivalence classes.

Consider the decomposition of X into its R-connected components with $R<\epsilon $ and the action of $\Gamma $ on these equivalence classes induced from $\Gamma \curvearrowright X$ .

The decomposition described above is a disjoint cover of X. If we repeat all of this again with $\epsilon ^{\prime }<\epsilon $ , the sets in the new cover will be contained inside the sets in the original. Building a system of covers this way and passing to the nerves (which are really just finite sets), we get that $\Gamma \curvearrowright X$ is conjugate (by way of the homeomorphism from X to the inverse limit of nerves) to an inverse limit of actions on the nerves where the equivariant connecting maps come from inclusions. Finally, if the original action was minimal, so is the odometer it is conjugate to, and an odometer is minimal if and only if the action on each level is transitive.

Proof. Suppose $\mu _{i}: \Gamma \to \text {Isom}(X)$ is as above. Then each $\mu _{i}$ gives an isometric action $F\curvearrowright X$ by the free group on S, where S is a finite generating set of $\Gamma $ . However, this action is conjugate to an inverse limit of $\Gamma $ -actions on finite sets, and so gives actions $\alpha _{N, i}$ of F on $E_{n}$ for each finite set $E_{n}$ . For any fixed N, we can then find $i_{N}$ sufficiently large so that $\alpha _{N, i}$ descends to an honest action $\Gamma \curvearrowright E_{N}$ .

Let $j_{N}: {\lim _{n\leftarrow } E_{n}}\to E_{N}$ be the usual projection. Then the pullbacks $j_{N}^{-1}(\{e\})$ for $e\in E_{N}$ are all homeomorphic to each other. So we can define an action $\Gamma \curvearrowright X\cong {\lim _{n\leftarrow } E_{n}}$ by homeomorphisms (just by fixing an identification ${\lim _{\leftarrow }}E_{n} \cong E_{N}\times \mathcal {C}$ and acting by $\alpha _{N}\times \text {id}_{\mathcal {C}}$ , where $\mathcal {C}$ is a Cantor set). Call this action $\tilde {\mu }_{i}$ . Then for $i\geq i_{N}$ , $\tilde {\mu }_{i}$ will be will be within $\epsilon _{N}$ of $\mu _{i}$ for some $\epsilon _{N}\to 0$ as $N\to \infty $ .

Proof. Let $g\in \text {Isom}(T^{n})$ . Let $h\in Isom(T^{n})$ be given by $h(x) = g(x) - g(0)$ , so $h(0) = 0$ . Since h is basepoint-preserving, it lifts to an isometry $\tilde {h}$ of $\mathbb {R}^{n}$ that fixes $0\in \mathbb {R}^{n}$ , that is, an element of $O_{n}(\mathbb {R})$ . As $\tilde {h}$ lifts a map of $T^{n}$ , $\tilde {h}(L) = L$ , so $\tilde {h}\in O_{n}(\mathbb {R})\cap \text {Aut}(L)$ which is a finite group F. Hence, h acts on $T^{n}$ via an element of F.

Since we can decompose isometries of $T^{n}$ as an element of F multiplied by a translation (as we did with g), there is a surjective map $F\times T^{n}\to \text {Isom}(T^{n})$ given by $(f, z)\mapsto (x\mapsto f\cdot x + z)$ . It is now straightforward to check that this map is injective, continuous, and a homomorphism for the semidirect product group structure on $F\ltimes T^{n}$ .

Proof. Let $\delta>0$ . We can assume S has the form $F\cup Z$ , where F is the finite part of the semidirect product forming $\text {Isom}(T^{n})$ as above and Z is a finite collection of rotations each in a single coordinate. Find N so that each irrational rotation in Z can be approximated to within $\delta $ by rational rotations with denominator N, and so that $\|b_{i}/N\|<\delta $ for all elements $b_{i}$ of some fixed basis of the lattice used to construct $T^{n}$ . Also make sure N contains as a factor the denominator of any rational rotations in Z (so that our approximation can be exact on any such rotation). Then each of these rational rotations acts on a set $E^{n}$ from the previous lemma, giving a $\delta $ -approximating action (for the generators in Z) on an $n\sqrt {n}\delta $ -dense subset of $T^{n}$ , which in fact coincides with the usual action on $E^{n}$ by some rational rotations inside $T^{n}$ .

The previous lemma now allows us to create an approximating action for the rest of $\langle S\rangle $ which is implemented through a translation action of $F\ltimes (\oplus _{k=1}^{n} \mathbb {Z}/N) < F\ltimes T^{n}$ , as it comes from F and rational rotations with denominator N. This action therefore respects the relations on the semi-direct product. Letting $\delta $ be sufficiently small compared to $\epsilon $ finishes the proof.

Proof. We can represent G as a subgroup of $U_{n}(\mathbb {C})$ for some n. Since the tangent spaces of $U_{n}(\mathbb {C})$ can be identified with smoothly varying subspaces of $M_{n}(\mathbb {C})$ , we can endow $U_{n}(\mathbb {C})$ with a Riemannian metric coming from the real part of the Hilbert–Schmidt inner product, and this Riemannian metric is invariant with respect to multiplication by elements of $U_{n}(\mathbb {C})$ .

Identify a set F of free-abelian generators for a finite index subgroup of $\Gamma $ isomorphic to $\mathbb {Z}^{k}$ . Since these elements commute with each other, we can conjugate by an appropriate unitary so that the matrices representing them are diagonal and so the closure $\overline {\mathbb {Z}^{k}}<G<U_{n}(\mathbb {C})$ is a torus. Restricting the aforementioned Riemannian metric gives a Riemannian metric on this torus which is invariant and therefore lifts to $\mathbb {R}^{k}$ . So this torus is constructed from a lattice like the tori considered in the previous lemmas.

Thus, if $x\in G$ , the closure of $\Gamma x$ in $G<U_{n}(\mathbb {C})$ is a finite disjoint union of isometric copies of that same torus (one for each of the finitely many cosets). Then as $\Gamma $ acts on $\overline {\Gamma x}<G$ by isometries, this action can be understood as permutations of these tori where each is transported isometrically into its image. So if we take a copy of $E^{n}$ as in the previous lemma inside each of these tori, we can create an approximating action for $\Gamma \curvearrowright \overline {\Gamma x}$ , as any element of $\Gamma $ will simply permute the copies of $E^{n}$ while applying something in $F\ltimes T^{n}$ to each of them. Since our choice of x at the beginning of this paragraph was arbitrary, we can repeat this process for different x as necessary to create an approximating action for $\Gamma \curvearrowright G$ .

Proof. The representation theory of compact groups shows that every compact group is an inverse limit of compact Lie groups (see, for instance, [Reference Dixmier2, Ch. 16]). We can assume further that the connecting maps are surjective. Moreover, every finitely generated, amenable subgroup of a compact Lie group is virtually abelian. This comes from combining the Tits alternative, Lie’s theorem, and Engel’s theorem. See, for instance, [Reference Yuan9].

So we have that the action $\Gamma \curvearrowright G$ is conjugate to $\Gamma \curvearrowright {\lim _{\leftarrow } G_{i}}$ , where $G_{i}$ are compact Lie groups, and the latter action comes from an action of $\Gamma $ on each $G_{i}$ via a map $\Gamma \to G_{i}$ so, in particular, the action on each $G_{i}$ is implemented by an amenable (and hence virtually abelian) subgroup of a compact Lie group. By the previous lemma, each action $\Gamma \curvearrowright G_{i}$ is therefore residually finite. We will show this implies residual finiteness of $\Gamma \curvearrowright {\lim _{\leftarrow } G_{i}}$ , which will finish the proof.

The pullbacks of open sets in the $G_{i}$ form a basis for the topology on G, so for any neighborhood of the diagonal in $G\times G$ , we can find a neighborhood contained in it which is a finite union of products of pullbacks of balls in various $G_{i}$ groups. Then we can find a K and $\delta>0$ such that if $(g_{i})$ and $(h_{i})$ are nets in the limit with $d(g_{i}, h_{i})<\delta $ for $i\leq K$ , then $((g_{i}), (h_{i}))$ is in this neighborhood of the diagonal in $G\times G$ .

Now find a $(\delta , F)$ -approximating action $\Gamma \curvearrowright E\subset G_{K}$ and embed E inside $\lim _{\leftarrow } G_{i}$ by taking any choice of pullbacks. This gives a residue for the given neighborhood of the diagonal.

Proof. As before, we can assume $\Gamma \curvearrowright X$ is isometric because of Remark 1.2.

Fix a finite generating set F. Since the action is isometric, we can identify $\Gamma $ with a subgroup of the compact group $\text {Isom}(X)$ . The action $\Gamma \curvearrowright G := \text {Isom}(X)$ by translations is then residually finite. Consider, for the uniform metric on $\text {Isom}(X)$ , an $(\epsilon , F)$ -residue $\Gamma \curvearrowright E$ where E contains the identity.

It suffices to verify the $\epsilon $ -density condition locally, so pick any $x\in X$ , and consider the orbit $Gx$ and the orbit map $\zeta : G\to Gx$ . Then for any $e_{0}\in E$ and $f\in F$ , we have $f\cdot e_{0} = e$ for some $e\in E$ with $d(e, f\cdot e_{0})<\epsilon $ , so $d(\zeta (f\cdot e_{0}), f\cdot \zeta (e_{0})) = d(ee_{0}\cdot x, fe_{0}\cdot x)<\epsilon $ , meaning $(Ex, \zeta )$ is an $(\epsilon , F)$ -residue for $\Gamma \curvearrowright X$ .