2.1 Basic concepts

Assume that ${\mathfrak {X}}$ is a Cantor space. Let $\mathrm {CO}({\mathfrak {X}})$ denote the collection of all clopen (closed and open) subsets of ${\mathfrak {X}}$ , which forms a basis for the topology of ${\mathfrak {X}}$ . For $\phi \in \mathrm {Homeo}({\mathfrak {X}})$ and $U \in \mathrm {CO}({\mathfrak {X}})$ , the image $\phi (U) \in \mathrm {CO}({\mathfrak {X}})$ . Recall that the action $({\mathfrak {X}},\Gamma ,\Phi )$ is minimal if for all $x \in {\mathfrak {X}}$ , its orbit ${\mathcal O}(x) = \{g \cdot x \mid g \in \Gamma \}$ is dense in ${\mathfrak {X}}$ .

The following result is folklore and a proof is given in [Reference Hurder and Lukina25, Proposition 3.1].

That is, given a clopen set $U \in \mathrm {CO}({\mathfrak {X}})$ , $\Gamma $ acts on the orbits of U by finite permutations. This justifies saying that a minimal equicontinuous Cantor action of $\Gamma $ is a (generalized) odometer.

We say that $U \subset {\mathfrak {X}}$ is adapted to the action $({\mathfrak {X}},\Gamma ,\Phi )$ if U is a non-empty clopen subset, and for any $g \in \Gamma $ , if $\Phi (g)(U) \cap U \ne \emptyset $ implies that $\Phi (g)(U) = U$ . Given $x \in {\mathfrak {X}}$ and clopen set $x \in W$ , there is an adapted clopen set U with $x \in U \subset W$ . (For example, see the proof of [Reference Hurder and Lukina25, Proposition 3.1].)

For an adapted set U, the set of ‘return times’ to U,

(2.1) $$ \begin{align} \Gamma_U = \{g \in \Gamma \mid g \cdot U \cap U \ne \emptyset \}, \end{align} $$

is a subgroup of $\Gamma $ , called the stabilizer of U. Then, for $g, g' \in \Gamma $ with ${g \cdot U \cap g' \cdot U \ne \emptyset} $ , we have $g^{-1} \, g' \cdot U = U$ , and hence $g^{-1} \, g' \in \Gamma _U$ . Thus, the translates $\{ g \cdot U \mid g \in \Gamma \}$ form a finite clopen partition of ${\mathfrak {X}}$ and are in 1–1 correspondence with the quotient space $X_U = \Gamma /\Gamma _U$ . Then, $\Gamma $ acts by permutations of the finite set $X_U$ and so the stabilizer group $\Gamma _U \subset G$ has finite index. Note that this implies that if $V \subset U$ is a proper inclusion of adapted sets, then the inclusion $\Gamma _V \subset \Gamma _U$ is also proper.

Given $x \in {\mathfrak {X}}$ and ${\varepsilon }> 0$ , Proposition 2.1 implies there exists an adapted clopen set ${U \in \mathrm {CO}({\mathfrak {X}})}$ with $x \in U$ and $\mathrm {diam}(U) < {\varepsilon }$ . Thus, one can choose a descending chain ${\mathcal U}$ of adapted sets in $\mathrm {CO}({\mathfrak {X}})$ whose intersection is x, from which we obtain the following result.

2.2 Equivalence of Cantor actions

We next recall the notions of equivalence of Cantor actions used in this work. The first and strongest is that of isomorphism, which is a generalization of the usual notion of conjugacy of topological actions. The definition below agrees with the usage in the papers [Reference Cortez and Medynets13, Reference Hurder and Lukina25, Reference Li31].

Definition 2.5. Cantor actions $({\mathfrak {X}}_1, \Gamma _1, \Phi _1)$ and $({\mathfrak {X}}_2, \Gamma _2, \Phi _2)$ are said to be isomorphic if there is a homeomorphism $h \colon {\mathfrak {X}}_1 \to {\mathfrak {X}}_2$ and group isomorphism $\Theta \colon \Gamma _1 \to \Gamma _2$ so that

(2.2) $$ \begin{align} \Phi_1(g) = h^{-1} \circ \Phi_2(\Theta(g)) \circ h \in \mathrm{Homeo}({\mathfrak{X}}_1) \quad \textrm{for all } g \in \Gamma_1. \end{align} $$

The notion of return equivalence for odometers is weaker than the notion of isomorphism, and is natural when considering the odometers defined by the holonomy actions for solenoidal manifolds, as considered in the works [Reference Hurder and Lukina24–Reference Hurder and Lukina26] and later in §4.

For an odometer $({\mathfrak {X}}, \Gamma , \Phi )$ and an adapted set $U \subset {\mathfrak {X}}$ , by an abuse of notation, we use $\Phi _U$ to denote both the restricted action $\Phi _U \colon \Gamma _U \times U \to U$ and the induced quotient action $\Phi _U \colon {\mathcal H}_U \times U \to U$ , where ${\mathcal H}_U = \Phi (\Gamma _U) \subset \mathrm {Homeo}(U)$ . Then, $(U, {\mathcal H}_U, \Phi _U)$ is called the restricted holonomy action for $\Phi $ , in analogy with the case where U is a transversal to a solenoidal manifold, and ${\mathcal H}_U$ is the holonomy group for this transversal. A technical issue that often arises though is that while $\Gamma _U \subset \Gamma $ has finite index, the action map $\Phi _U \colon \Gamma _U \to {\mathcal H}_U$ need not be injective, and can in fact can have a large kernel as is the case, for example, for the actions of weakly branch groups (see Example 6.10).

Note that if we take $U = {\mathfrak {X}}$ and $U' = {\mathfrak {X}}'$ in Definition 2.6, then return equivalence may still be weaker than isomorphism in Definition 2.5, unless the actions $\Phi $ and $\Phi '$ are topologically free [Reference Hurder and Lukina25, Reference Li31, Reference Renault37].

2.3 Algebraic Cantor actions

There is a natural basepoint $x_{\infty } \in X_{\infty }$ given by the cosets of the identity element $e \in \Gamma $ , so $x_{\infty } = (e \Gamma _{\ell })$ . An adapted neighborhood basis of $x_{\infty }$ is given by the clopen sets

(2.3) $$ \begin{align} U_{\ell} = \{ x = (x_{i}) \in X_{\infty} \mid x_i = e \Gamma_i \in X_i, 0 \leq i \leq \ell ~ \} \subset X_{\infty}. \end{align} $$

Then, there is the tautological identity $\Gamma _{\ell } = \Gamma _{U_{\ell }}$ .

Suppose that we are given an odometer $({\mathfrak {X}},\Gamma ,\Phi )$ and an adapted neighborhood basis ${\mathcal U}$ . Define subgroups $\Gamma _{\ell } = \Gamma _{U_{\ell }}$ , with $\Gamma _0 = \Gamma $ , which form the group chain $ {\mathcal G}_{{\mathcal U}} = \{\Gamma _0 \supset \Gamma _1 \supset \cdots \}$ . Then we have the following folklore result.

3.1 Types and typesets

Recall that a Steinitz number $\xi $ can be written uniquely as the formal product over the set of primes,

(3.1) $$ \begin{align} \xi = \prod_{p \in \Pi} ~ p^{\chi_{\xi}(p)} , \end{align} $$

where the characteristic function $\chi _{\xi } \colon \Pi \to \{0,1,\ldots , \infty \}$ counts the multiplicity with which a prime p appears in the infinite product $\xi $ . Note that multiplication of Steinitz numbers corresponds to the sum of their characteristic functions.

Recall from Definition 1.2 that two Steinitz numbers $\xi $ and $\xi '$ are said to be asymptotically equivalent if there exists finite integers $m, m' \geq 1$ such that $m \cdot \xi = m' \cdot \xi '$ , and we then write . Recall that the type $\tau [\xi ]$ associated to a Steinitz number $\xi $ is the asymptotic equivalence class of $\xi $ .

Given two types $\tau $ and $\tau '$ , we write $\tau \leq \tau '$ if there exists representatives $\xi \in \tau $ and ${\xi ' \in \tau '}$ such that their characteristic functions satisfy $\chi _{\xi }(p) \leq \chi _{\xi '}(p)$ for all primes $p \in \Pi $ . Then, two Steinitz numbers $\xi $ and $\xi '$ are asymptotically equivalent if and only if $\chi _{\xi } \leq \chi _{\xi '}$ and $\chi _{\xi '} \leq \chi _{\xi }$ .

There are three operations on types $\tau $ and $\tau '$ : product, join, and intersection. Let $\chi $ (respectively $\chi '$ ) be the characteristic function for a representative ${\xi \in \tau} $ (respectively $\xi ' \in \tau '$ ), then

$$ \begin{align*} \text{Product: } \tau \cdot \tau' \text{ type defined by } \chi^{+}(p) & = \chi(p) + \chi'(p); \\ \text{Join: } \tau \vee \tau' \text{ type defined by } \chi^{\vee}(p)& = \max \{ \chi(p) , \chi'(p)\}; \\ \text{Intersection: } \tau \wedge \tau' \text{ type defined by } \chi^{\wedge}(p) & = \min \{ \chi(p) , \chi'(p)\}. \end{align*} $$

Note that $\tau \wedge \tau ' \leq \tau \vee \tau ' \leq \tau \cdot \tau '$ . A typeset $\Xi $ need not be closed under the operations of product, join, or intersection. However, a typeset $\Xi $ always admits a partial ordering.

3.2 Type for profinite groups

A topological group ${\mathfrak {G}}$ is said to be profinite if it is isomorphic to the inverse limit of finite groups (see [Reference Wilson48, Ch. 2] or [Reference Ribes and Zalesskii39, Ch. 2.3]).

The Steinitz order $\Pi [{\mathfrak {G}}]$ of a profinite group ${\mathfrak {G}}$ is defined by the supernatural number associated to a presentation of ${\mathfrak {G}}$ , defined as follows. For a profinite group ${\mathfrak {G}}$ , an open subgroup ${\mathfrak {U}} \subset {\mathfrak {G}}$ has finite index [Reference Ribes and Zalesskii39, Lemma 2.1.2]. Let ${\mathfrak {D}} \subset {\mathfrak {G}}$ be a closed subgroup and ${\mathfrak {N}} \subset {\mathfrak {G}}$ is an open normal subgroup, then ${\mathfrak {N}} \cdot {\mathfrak {D}}$ is an open subgroup of ${\mathfrak {G}}$ and ${\mathfrak {N}} \cap {\mathfrak {D}}$ is an open normal subgroup of ${\mathfrak {D}}$ .

The Steinitz number $\xi ({\mathfrak {G}} : {\mathfrak {D}})$ is called the relative Steinitz order of the pair $({\mathfrak {G}}, {\mathfrak {D}} )$ .

The Steinitz orders satisfy the Lagrange identity, where the multiplication is taken in the sense of supernatural numbers (see [Reference Ribes and Zalesskii39, Reference Wilson48]), and we have

(3.2) $$ \begin{align} \xi({\mathfrak{G}}) = \xi({\mathfrak{G}} : {\mathfrak{D}}) \cdot \xi({\mathfrak{D}}). \end{align} $$

In particular, we always have $\xi [{\mathfrak {D}}] \leq \xi [{\mathfrak {G}}]$ .

3.3 Type for odometers

Let $(X_{\infty }, \Gamma , \Phi _{\infty })$ be an odometer, defined by a group chain ${\mathcal G} = \{\Gamma = \Gamma _0 \supset \Gamma _1 \supset \cdots \}$ . Recall that the normal core of $\Gamma _{\ell }$ is the largest normal subgroup $C_{\ell } \subset \Gamma _{\ell }$ . The profinite group $\widehat {\Gamma }({\mathcal G})$ associated to the action $(X_{\infty }, \Gamma , \Phi _{\infty })$ defined by ${\mathcal G}$ is given by the following definition.

Definition 3.5. Let ${\mathcal G}$ be a group chain with associated normal core chain $\{C_{\ell } \mid \ell \geq 0\}$ , then

(3.3) $$ \begin{align} \widehat{\Gamma}({\mathcal G}) = \lim_{\longleftarrow} \{ \Gamma/C_{\ell +1} \to \Gamma/C_{\ell} \mid \ell \geq 0 \}. \end{align} $$

The profinite group $\widehat {\Gamma }({\mathcal G})$ is a quotient of the full profinite completion of $\Gamma $ , but it is typically not equal to the profinite completion. The properties of the group $\widehat {\Gamma }({\mathcal G})$ associated to a Cantor action are studied extensively in the authors’ works [Reference Dyer, Hurder and Lukina17, Reference Dyer, Hurder and Lukina18, Reference Hurder and Lukina24, Reference Hurder and Lukina26]. The Steinitz order of $\widehat {\Gamma }({\mathcal G})$ is well defined and given by

(3.4) $$ \begin{align} \xi(\widehat{\Gamma}({\mathcal G})) = \mathrm{lcm} \{ \# ( \Gamma/C_{\ell}) \mid \ell> 0 \}. \end{align} $$

The type $\tau [\widehat {\Gamma }({\mathcal G})] = \tau [\xi (\widehat {\Gamma }({\mathcal G}))]$ need not be an invariant of return equivalence of the odometer $(X_{\infty }, \Gamma , \Phi _{\infty })$ , as explained below.

Definition 3.6. Let $(X_{\infty }, \Gamma , \Phi _{\infty })$ be an odometer. The type $\tau [X_{\infty }, \Gamma , \Phi _{\infty }]$ of the action is the equivalence class of the Steinitz number

(3.5) $$ \begin{align} \xi(X_{\infty}, \Gamma, \Phi_{\infty}) = \mathrm{lcm} \{ \# X_{\ell} = \#(\Gamma/\Gamma_{\ell}) \mid \ell> 0 \}. \end{align} $$

Recall that there is a transitive action $\widehat {\Phi }_{\infty } \colon \widehat {\Gamma }({\mathcal G}) \times X_{\infty } \to X_{\infty }$ induced by odometer $(X_{\infty }, \Gamma , \Phi _{\infty })$ . The action $(X_{\infty }, \widehat {\Gamma }({\mathcal G}), \widehat {\Phi }_{\infty })$ is free precisely when the isotropy subgroup ${\mathcal D}({\mathcal G}) \subset \widehat {\Gamma }({\mathcal G})$ of the action at the basepoint $x_{\infty } \in X_{\infty }$ is trivial. In this case, we have a homeomorphism $X_{\infty } \cong \widehat {\Gamma }({\mathcal G})$ that commutes with the action, and so $X_{\infty }$ inherits the structure of a Cantor group from the action.

However, when the action $(X_{\infty }, \widehat {\Gamma }({\mathcal G}), \widehat {\Phi }_{\infty })$ is not free, then ${\mathfrak {D}}({\mathcal G})$ is not trivial and we have the following proposition.

We omit the proof of this, as it is a direct consequence of the definitions, and the result is not needed for the proofs of our main theorems. However, Proposition 3.7 provides some insights on the properties of the type $\tau [X_{\infty }, \Gamma , \Phi _{\infty }]$ of an odometer with respect to return equivalence.

The Lagrange theorem for profinite groups, equation (3.2), implies that $\tau [\widehat {\Gamma }({\mathcal G})] = \tau [X_{\infty }, \Gamma , \Phi _{\infty }] \cdot \tau [{\mathcal D}({\mathcal G})]$ . The type $\tau [{\mathcal D}({\mathcal G})]$ need not be invariant under restriction to adapted subsets, and so $\tau [{\mathcal D}({\mathcal G})]$ need not be invariant under the relation of return equivalence, and thus the same is true for $\tau [\widehat {\Gamma }({\mathcal G})]$ . However, the proof of Theorem 1.5 shows that the relative type $\tau [\widehat {\Gamma }({\mathcal G}) : {\mathcal D}({\mathcal G})] = \tau [{\mathcal G}]$ is invariant under restriction to adapted subsets.

3.4 Typesets for odometers

We next consider the properties of typeset under restriction, which leads to the notion of commensurable typesets. Let ${\mathcal G} = \{\Gamma = \Gamma _0 \supset \Gamma _1 \supset \Gamma _2 \supset \cdots \}$ be a group chain and $(X_{\infty }, \Gamma , \Phi _{\infty })$ the associated odometer.

Let $H \subset \Gamma $ be a subgroup of finite index such that $\Gamma _{\ell } \subset H$ for some $\ell> 0$ . By omitting some initial terms, we can assume that $\Gamma _1 \subset H$ . Then, for each $\ell> 0$ , define

(3.6) $$ \begin{align} C_{\ell}^H = \bigcap_{\delta \in H} ~ \delta \Gamma_{\ell} \delta^{-1}. \end{align} $$

Then, $C_{\ell }^H \subset \Gamma _{\ell }$ is a subgroup of finite index, as $\Gamma _{\ell }$ has finite index in $\Gamma $ . In particular, $C_{\ell } = C_{\ell }^{\Gamma }$ is the normal core of $\Gamma _{\ell }$ and there is an inclusion $C_{\ell } \subset C_{\ell }^H$ for any choice of H. Also note that $C_{\ell +1}^H \subset C_{\ell }^H$ for all $\ell \geq 0$ .

Given $\gamma \in \Gamma $ , define $C_{\gamma , \ell }^H = \langle \gamma \rangle \cap C_{\ell }^H$ , and let ${\mathcal C}_{\gamma }^H = \{\langle \gamma \rangle \cap H = C_{\gamma , 0}^H \supset C_{\gamma , 1}^H \supset C_{\gamma , 2}^H \supset \cdots \}$ denote the resulting subgroup chain in $\langle \gamma \rangle $ .

Definition 3.8. Given a group chain ${\mathcal G}$ and $H \subset \Gamma $ a subgroup with $\Gamma _{\ell } \subset H$ for some $\ell \geq 1$ , the H-restricted order of $\gamma \in \Gamma $ is the Steinitz order with respect to the chain ${\mathcal C}_{\gamma }^H$ ,

(3.7) $$ \begin{align} \xi^H(\gamma) = \mathrm{lcm} \{ \#( \langle \gamma \rangle /C_{\gamma, \ell}^H ) \mid \ell> 0 \}. \end{align} $$

The H-restricted typeset for the chain ${\mathcal G}$ is the collection

(3.8) $$ \begin{align} \Xi_H[{\mathcal G}] = \{ \tau[\xi^H(\gamma)] \mid \gamma \in \Gamma\}. \end{align} $$

Note that when $H = \Gamma $ , we recover the typeset $\Xi [{\mathcal G}]$ in Definition 1.6.

We can now formulate the notion of commensurable typesets for group chains ${\mathcal G}$ and ${\mathcal G}'$ .

Finally, we give an elementary result about the types of elements $\gamma \in \Gamma $ that follows from basic group theory. For all $\ell> 0$ , we have that $\langle \gamma \rangle _{\ell } \subset C_{\ell }$ , so the order of the subgroup $\langle \gamma \rangle /\langle \gamma \rangle _{\ell }$ divides the order of $\Gamma /C_{\ell }$ by Lagrange’s theorem. Thus, we have the following proposition.

4.1 Proof of Theorem 1.5

We must show that the Steinitz number $\xi ({\mathfrak {X}},\Gamma ,\Phi )$ of an odometer $({\mathfrak {X}},\Gamma ,\Phi )$ is well defined and that its type $\tau [\xi ({\mathfrak {X}},\Gamma ,\Phi )]$ is an invariant under return equivalence. The first claim is a special case of the proof of invariance under return equivalence, as explained later.

Suppose that the odometers $({\mathfrak {X}}, \Gamma , \Phi )$ and $({\mathfrak {X}'}, \Gamma ', \Phi ')$ are return equivalent. Then, by Theorem 2.7, the odometers $(X_{\infty }, \Gamma , \Phi _{\infty })$ and $(X_{\infty }', \Gamma ', \Phi _{\infty }')$ are return equivalent, as in Definition 2.6.

Choose adapted sets $U \subset X_{\infty }$ and $U' \subset X^{\prime }_{\infty }$ and a homeomorphism $h \colon U \to U'$ so that the induced homomorphism $h_* \colon \mathrm {Homeo}(U) \to \mathrm {Homeo}(U')$ restricts to an isomorphism between the image ${\mathcal H}_U = \Phi (\Gamma _U)$ with the image ${\mathcal H}^{\prime }_{U'} = \Phi '(\Gamma ^{\prime }_{U'})$ .

Recall a standard construction of a diagram of maps between adapted sets [Reference Fokkink and Oversteegen20], [Reference Dyer, Hurder and Lukina17, Theorem 3.3]:

(4.1)

The sets $U_{\ell }$ are defined in equation (2.3) and adapted to the action of $\Gamma $ , and similarly for the sets $U^{\prime }_{\ell '}$ , which are adapted to the action of $\Gamma '$ . The subscripts and intermediate adapted sets are defined iteratively in the following. We adopt the ‘ $\cdot $ ’ notation for the actions, as it is clear from the context which action is being applied. Recall that the action of $\Gamma _U$ on U is minimal, as is the action of $\Gamma ^{\prime }_{U'}$ on $U'$ . Denote $e_{\infty } = (e\Gamma _\ell ) \in X_{\infty }$ and $e_{\infty }^{\prime } = (e \Gamma _{\ell '}^{\prime }) \in X_{\infty }^{\prime }$ , where $\Gamma _\ell $ is the isotropy subgroup of $U_\ell $ and $\Gamma _{\ell }^{\prime }$ is the isotropy subgroup of $U_{\ell '}^{\prime }$ , for each $\ell \geq 0$ and each $\ell ' \geq 0$ .

We define the maps and indices in diagram (4.1) recursively.

The set U is clopen, so there exists $\gamma _1 \in \Gamma $ such that $\gamma _1 \cdot e_{\infty } \in U$ . Choose $\ell _1> 0$ such that $\gamma _1 \cdot U_{\ell _1} \subset U$ .

The image $h(\gamma _1 \cdot U_{\ell _1}) \subset U'$ is clopen, so choose $\gamma ^{\prime }_1 \in \Gamma '$ such that $\gamma ^{\prime }_1 \cdot e^{\prime }_{\infty } \in h(\gamma _1 \cdot U_{\ell _1})$ . Choose $\ell ^{\prime }_1> 0$ such that $\gamma ^{\prime }_1 \cdot U^{\prime }_{\ell ^{\prime }_1} \subset h(\gamma _1 \cdot U_{\ell _1})$ .

The image $h^{-1}(\gamma ^{\prime }_1 \cdot U^{\prime }_{\ell ^{\prime }_1}) \subset \gamma _1 \cdot U_{\ell _1}$ is clopen, so choose $\gamma _2 \in \Gamma $ such that $\gamma _2 \cdot e_{\infty } \in h^{-1}(\gamma ^{\prime }_1 \cdot U^{\prime }_{\ell ^{\prime }_1}) \subset \gamma _1 \cdot U_{\ell _1}$ . Choose $\ell _2> \ell _1$ such that $\gamma _2 \cdot U_{\ell _2} \subset h^{-1}(\gamma ^{\prime }_1 \cdot U^{\prime }_{\ell ^{\prime }_1})$ .

The image $h(\gamma _2 \cdot U_{\ell _2}) \subset \gamma ^{\prime }_1 \cdot U^{\prime }_{\ell ^{\prime }_1}$ is clopen, so choose $\gamma ^{\prime }_2 \in \Gamma '$ such that $\gamma ^{\prime }_2 \cdot e^{\prime }_{\infty } \in h(\gamma _2 \cdot U_{\ell _2})$ . Choose $\ell ^{\prime }_2> \ell ^{\prime }_1$ such that $\gamma ^{\prime }_2 \cdot U^{\prime }_{\ell ^{\prime }_2} \subset h(\gamma _2 \cdot U_{\ell _2})$ .

Continue this procedure recursively to obtain diagram (4.1), where we have:

• • increasing sequences $0< \ell _1 < \ell _2 < \ell _3 < \cdots $ and $0 < \ell ^{\prime }_1 < \ell ^{\prime }_2 < \ell ^{\prime }_3 < \cdots $ ;

• • a sequence $\{\gamma _{1}, \gamma _{2}, \gamma _{3}, \ldots \} \subset \Gamma $ ;

• • a sequence $\{\gamma ^{\prime }_{1}, \gamma ^{\prime }_{2}, \gamma ^{\prime }_{3}, \ldots \} \subset \Gamma '$ .

Observe that, by choice, we have $\gamma _{i+1} \cdot U_{\ell _{i+1}} \subset \gamma _{i} \cdot U_{\ell _i}$ and thus $\gamma _{\ell _i}^{-1} \gamma _{\ell _{i+1}} \in \Gamma _{\ell _i}$ for all ${i \geq 1}$ . This recursion relation implies that the sequence $\{\gamma _i \mid i \geq 0\}$ converges in the profinite topology on $\Gamma $ induced by the subgroup chain $\{\Gamma _{\ell } \mid \ell \geq 0\}$ , and similarly for $\{\gamma _i' \mid i \geq 0\}$ in the profinite topology on $\Gamma '$ . If we denote the respective limits by $\overline {\{\gamma _i\}} \in \overline {\Gamma }$ and $\overline {\{\gamma _i'\}} \in \overline {\Gamma }'$ , then we have

$$ \begin{align*}\overline{\{\gamma_i\}} \cdot e_{\infty} = h^{-1}(\overline{\{\gamma_i'\}} \cdot e^{\prime}_{\infty}) \quad\textrm{and}\quad\overline{\{\gamma^{\prime}_i\}} \cdot e^{\prime}_{\infty} = h(\overline{\{\gamma_i\}} \cdot e_{\infty}).\end{align*} $$

All of the sets appearing in diagram (4.1) are adapted for their respective actions, so the set inclusions induce corresponding subgroup chains of their isotropy groups, and these chains are interlaced:

(4.2)

Conjugation does not change the index of a subgroup, so $[\Gamma : \Gamma _{\ell _j}] = [\Gamma : \gamma _j \Gamma _{\ell _j} \gamma _j^{-1}]$ for all $j \geq 1$ , and likewise we have $[\Gamma ' : \Gamma ^{\prime }_{\ell ^{\prime }_j}] = [\Gamma ' : \gamma ^{\prime }_j \Gamma ^{\prime }_{\ell ^{\prime }_j} (\gamma ^{\prime }_j)^{-1}]$ . It then follows that

(4.3) $$ \begin{align} [\Gamma : \Gamma_{\ell_j}] = [\Gamma : \Gamma_{\ell_1} ] [\Gamma_{\ell_1}: \Gamma_{\ell_j}] = [\Gamma : \Gamma_{\ell_1} ][\gamma_1 \Gamma_{\ell_1} \gamma_1^{-1} : \gamma_j \Gamma_{\ell_j} \gamma_j^{-1}] , \end{align} $$

(4.4) $$ \begin{align} [\Gamma' : \Gamma^{\prime}_{\ell^{\prime}_j}] = [ \Gamma' : \Gamma^{\prime}_{\ell^{\prime}_1} ] [\Gamma^{\prime}_{\ell^{\prime}_1} : \Gamma^{\prime}_{\ell^{\prime}_j} ] = [ \Gamma' : \Gamma^{\prime}_{\ell_1}][\gamma^{\prime}_1 \Gamma^{\prime}_{\ell^{\prime}_1} (\gamma^{\prime}_1)^{-1} : \gamma^{\prime}_j \Gamma^{\prime}_{\ell^{\prime}_j} (\gamma^{\prime}_j)^{-1}]. \end{align} $$

We now show the key fact for the proof of Theorem 1.5.

Lemma 4.1. For all $j> i > 0$ ,

(4.5) $$ \begin{align} [\Gamma_{\ell_i}: \Gamma_{\ell_j}] = [\Gamma^{\prime}_{h(\gamma_i \cdot U_{\ell_i})} : \Gamma^{\prime}_{h(\gamma_j \cdot U_{\ell_j})}]. \end{align} $$

We now prove that $\tau [\xi (X_{\infty }, \Gamma , \Phi _{\infty })] = \tau [\xi (X_{\infty }', \Gamma ', \Phi _{\infty }')]$ . Recall that

(4.6) $$ \begin{align} \xi({\mathcal G}_{\mathcal U}) = \mathrm{lcm} \{ [\Gamma : \Gamma_{\ell}] \mid \ell> 0\} = \mathrm{lcm} \{ [\Gamma : \Gamma_{\ell_j}] \mid j > 0 \} , \end{align} $$

(4.7) $$ \begin{align} \xi({\mathcal G}^{\prime}_{{\mathcal U}'}) = \mathrm{lcm} \{ [\Gamma' : \Gamma^{\prime}_{\ell}] \mid \ell> 0\} = \mathrm{lcm} \{ [\Gamma' : \Gamma^{\prime}_{\ell^{\prime}_j}] \mid j > 0 \}. \end{align} $$

Then, by equation (4.5), for $j> 1$ , we have

(4.8) $$ \begin{align} [ \Gamma : \Gamma_{\ell_j} ] = [ \Gamma : \Gamma_{\ell_1}] \ [\Gamma_{\ell_1} : \Gamma_{\ell_j}] = [ \Gamma : \Gamma_{\ell_1}] \ [\Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} : \Gamma^{\prime}_{h(\gamma_j \cdot U_{\ell_j})}]. \end{align} $$

Then calculate, recalling the inclusions $U' \supset h(\gamma _1 \cdot U_{\ell _1}) \supset \gamma _1' \cdot U_{\ell _1}' \supset h(\gamma _j \cdot U_{\ell _j}) \supset \gamma _j' \cdot U_{\ell _j'}'$ from equation (4.1):

(4.9) $$ \begin{align} &[\Gamma' : \Gamma^{\prime}_{\ell^{\prime}_j}] = [\Gamma' : \gamma_j'\Gamma^{\prime}_{\ell^{\prime}_j} (\gamma_j')^{-1}] \nonumber\\ & \ = [\Gamma' : \Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} ] \ [\Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} : \Gamma^{\prime}_{\gamma^{\prime}_1 \cdot U^{\prime}_{\ell^{\prime}_1}} ] \ [\Gamma^{\prime}_{\gamma^{\prime}_1 \cdot U^{\prime}_{\ell^{\prime}_1}} : \Gamma^{\prime}_{h(\gamma_{j} \cdot U_{\ell_{j}})}] \ [ \Gamma^{\prime}_{h(\gamma_{j} \cdot U_{\ell_{j}})} : \Gamma^{\prime}_{\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j}}] \nonumber \\ & \ = [\Gamma' : \Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} ] \ [\Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} : \Gamma^{\prime}_{h(\gamma_{j} \cdot U_{\ell_{j}})}] \ [\Gamma^{\prime}_{h(\gamma_{j} \cdot U_{\ell_{j}})} : \Gamma^{\prime}_{\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j}}]. \end{align} $$

Then, by equation (4.8), the last line of equation (4.9) yields

(4.10) $$ \begin{align} [\Gamma : \Gamma_{\ell_1}] \ [\Gamma' : \Gamma^{\prime}_{\ell^{\prime}_{j}}] = [\Gamma' : \Gamma^{\prime}_{h(\gamma_1 \cdot U_{\ell_1})} ] \ [ \Gamma : \Gamma_{\ell_j} ] \ [\Gamma^{\prime}_{h(\gamma_{j} \cdot U_{\ell_{j}})} : \Gamma^{\prime}_{\gamma^{\prime}_{j} \cdot U^{\prime}_{\ell^{\prime}_{j}}}]. \end{align} $$

Thus, the index $[ \Gamma : \Gamma _{\ell _j}]$ divides $[\Gamma : \Gamma _{\ell _1}] \ [\Gamma ' : \Gamma ^{\prime }_{\ell ^{\prime }_{j}}]$ for all $j> 1$ . In particular, ${[ \Gamma : \Gamma _{\ell _j}]}$ divides $[\Gamma : \Gamma _{\ell _1}] \ \xi ({\mathcal P}')$ for all $j> 1$ , and so $\xi ({\mathcal P})$ divides $[\Gamma : \Gamma _{\ell _1}] \ \xi ({\mathcal P}')$ ; hence $\tau [{\mathcal P}] \leq \tau [{\mathcal P}']$ .

Next, repeat these calculations for $j> 1$ , starting with

(4.11) $$ \begin{align} [ \Gamma : \Gamma_{\ell_{j+1}} ] = [ \Gamma : \Gamma_{h^{-1}(\gamma^{\prime}_1 \cdot U_{\ell^{\prime}_1})}] \ [\Gamma_{h^{-1}(\gamma^{\prime}_1 \cdot U_{\ell^{\prime}_1})} : \Gamma_{h^{-1}(\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j})}] \ [\Gamma_{h^{-1}(\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j})} : \Gamma_{\gamma_{j+1} \cdot U_{\ell_{j+1}}}]. \end{align} $$

Lemma 4.1 can be applied to the inverse map $h^{-1} \colon U' \to U$ as well, to obtain that for all $j> i > 0$ ,

(4.12) $$ \begin{align} [\Gamma^{\prime}_{\ell^{\prime}_i}: \Gamma^{\prime}_{\ell^{\prime}_j}] = [\Gamma_{h^{-1}(\gamma^{\prime}_i \cdot U_{\ell^{\prime}_i})} : \Gamma_{h^{-1}(\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j})}] \end{align} $$

and so

(4.13) $$ \begin{align} [ \Gamma : \Gamma_{\ell_{j+1}} ] = [ \Gamma : \Gamma_{h^{-1}(\gamma^{\prime}_1 \cdot U_{\ell^{\prime}_1})}] \ [\Gamma^{\prime}_{\ell^{\prime}_1}: \Gamma^{\prime}_{\ell^{\prime}_j}] \ [\Gamma_{h^{-1}(\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j})} : \Gamma_{\gamma_{j+1} \cdot U_{\ell_{j+1}}}]. \end{align} $$

Thus, the index $[\Gamma ^{\prime }_{\ell ^{\prime }_1}: \Gamma ^{\prime }_{\ell ^{\prime }_j}]$ divides $[\Gamma : \Gamma _{\ell _{j+1}}]$ for all $j> 1$ and so $[\Gamma ^{\prime }_{\ell ^{\prime }_1}: \Gamma ^{\prime }_{\ell ^{\prime }_j}]$ divides $\xi ({\mathcal G}_{\mathcal U})$ . It follows that $\xi ({\mathcal G}^{\prime }_{{\mathcal U}'})$ divides $\xi ({\mathcal G}_{\mathcal U})$ ; hence, $\tau [{\mathcal G}^{\prime }_{{\mathcal U}'}] \leq \tau [{\mathcal G}_{\mathcal U}]$ . Reversing the roles of the group chains ${\mathcal G}_{\mathcal U}$ and ${\mathcal G}^{\prime }_{{\mathcal U}'}$ , we obtain that $\tau [{\mathcal G}_{\mathcal U}] = \tau [{\mathcal G}^{\prime }_{{\mathcal U}'}]$ , and thus $\tau [{\mathfrak {X}}, \Gamma , \Phi )] = \tau [{\mathfrak {X}'}, \Gamma ', \Phi ']$ . This completes the proof that the types of return equivalent actions are equal.

Finally, to complete the proof of Theorem 1.5, we show that $\xi ({\mathcal G}_{\mathcal U})= \xi ({\mathcal G}^{\prime }_{{\mathcal U}'})$ for group chains ${\mathcal G}_{\mathcal U}$ and ${\mathcal G}^{\prime }_{{\mathcal U}'}$ associated with $({\mathfrak {X}},\Gamma ,\Phi )$ . We proceed as in the above proof and note that while $\Gamma = \Gamma '$ , the group chains define distinct inverse limit spaces $X_{\infty }$ and $X^{\prime }_{\infty }$ . In the above calculations, take $U= X_{\infty }$ and $U'=X^{\prime }_{\infty }$ , $\gamma _1 = e \in \Gamma $ with $\ell _1 = 0$ , and $\gamma ^{\prime }_1 = e' \in \Gamma '$ with $\ell ^{\prime }_1 = 0$ . Then, the terms $ [\Gamma : \Gamma _{\ell _1}] =1$ and $[\Gamma ' : \Gamma ^{\prime }_{\ell ^{\prime }_1}] = 1$ . Then, by equation (4.10), we have $[ \Gamma : \Gamma _{\ell _j} ]$ divides $[\Gamma ' : \Gamma ^{\prime }_{\ell ^{\prime }_{j}}]$ and by equation (4.13), we have $[\Gamma ' : \Gamma ^{\prime }_{\ell ^{\prime }_j}]$ divides $[ \Gamma : \Gamma _{\ell _{j+1}} ]$ . It follows that $\xi ({\mathcal G}_{\mathcal U})= \xi ({\mathcal G}^{\prime }_{{\mathcal U}'})$ . That is, the Steinitz order is independent of the choice of a group associated to the action.

4.2 Proof of Theorem 1.8

We must show that the typesets of odometers are well defined and invariant under return equivalence modulo the equivalence relation on typesets in Definition 3.9. As above, we show the result for return equivalence first, then deduce the result for conjugation from the proof of this. The proof begins as in §4.1.

Suppose that the odometers $({\mathfrak {X}}, \Gamma , \Phi )$ and $({\mathfrak {X}'}, \Gamma ', \Phi ')$ are return equivalent, then the odometers $(X_{\infty }, \Gamma , \Phi _{\infty })$ and $(X_{\infty }', \Gamma ', \Phi _{\infty }')$ are return equivalent. Choose adapted sets $U \subset X_{\infty }$ and $U' \subset X^{\prime }_{\infty }$ , and a homeomorphism $h \colon U \to U'$ so that the induced homomorphism $h_* \colon \mathrm {Homeo}(U) \to \mathrm {Homeo}(U')$ restricts to an isomorphism between the image ${\mathcal H}_U = \Phi (\Gamma _U)$ with the image ${\mathcal H}^{\prime }_{U'} = \Phi '(\Gamma ^{\prime }_{U'})$ that induces an isomorphism between the actions $(U, {\mathcal H}_{U}, \Phi _{U})$ and $(U', {\mathcal H}^{\prime }_{U'}, \Phi ^{\prime }_{U'})$ .

Next, choose basepoints $x \in U$ and $x' \in U'$ . Choose an adapted neighborhood basis ${\mathcal U} = \{U_{\ell } \subset {\mathfrak {X}} \mid \ell> 0\}$ at x for the action $\Phi $ as in Definition 2.2, and an adapted neighborhood basis ${\mathcal U}' = \{U^{\prime }_{\ell } \subset {\mathfrak {X}}' \mid \ell> 0\}$ at $x'$ for the action $\Phi '$ . We can assume that $U_1 \subset U$ and $U^{\prime }_1 \subset U'$ .

Form the group chain $ {\mathcal G} = \{\Gamma _0 \supset \Gamma _1 \supset \cdots \}$ , where $\Gamma _{\ell } = \Gamma _{U_{\ell }}$ with $\Gamma _0 = \Gamma $ , and the group chain $ {\mathcal G}' = \{\Gamma ^{\prime }_0 \supset \Gamma ^{\prime }_1 \supset \cdots \}$ , where $\Gamma ^{\prime }_{\ell } = \Gamma ^{\prime }_{U^{\prime }_{\ell }}$ with $\Gamma ^{\prime }_0 = \Gamma '$ .

Set $H = \Gamma _U$ and $H' = \Gamma ^{\prime }_{U'}$ . We will show that the H-restricted typeset $\Xi _H[X_{\infty }, \Gamma , \Phi ]$ and the $H'$ -restricted typeset $\Xi _{H'}[X^{\prime }_{\infty } , \Gamma ' , \Phi ']$ are equal.

Given $\gamma \in \Gamma $ , for any positive integer $m> 0$ , the group $\langle \gamma ^m \rangle $ is a subgroup of finite index in $\langle \gamma \rangle $ . Thus, $\xi ^H(\gamma ) = m \xi ^H(\gamma ^m)$ and $\tau [\xi ^H(\gamma )] = \tau [\xi ^H(\gamma ^m)]$ . Since H has finite index in $\Gamma $ , for any $\gamma \in \Gamma $ , there exists $m> 0$ such that $\gamma ^m \in H$ . Thus, we have $\Xi _H[{\mathcal G}] = \Xi _H[H \cap {\mathcal G}]$ , so it suffices to consider $\gamma \in H$ , and likewise for $\gamma ' \in H'$ .

Let sequences $\{\gamma _i \mid i \geq 1\}$ , $\{\gamma ^{\prime }_i \mid i \geq 1\}$ , $\{\ell _i \mid i \geq 1\}$ and $\{\ell ^{\prime }_i \mid i \geq 1\}$ be chosen as in §4.1, resulting in diagram (4.1) of adapted sets, and diagram (4.2) of group inclusions. Note that as we assume $e_{\infty } \in U$ , we can choose $\gamma _1$ to be the identity, and likewise as $e^{\prime }_{\infty } \in U'$ , we choose $\gamma ^{\prime }_1$ to be the identity. By choice, we have $\gamma _{i+1} \cdot U_{\ell _{i+1}} \subset \gamma _{i} \cdot U_{\ell _i}$ and thus $\gamma _{\ell _i}^{-1} \gamma _{\ell _{i+1}} \in \Gamma _{\ell _i}$ for all $i \geq 1$ . In particular, this implies that $\gamma _i \in H$ for all $i \geq 1$ . The analogous conclusion holds, that $\gamma ^{\prime }_i \in H'$ for all $i \geq 1$ .

For notational convenience, set $H_i = \Gamma _{\gamma _i \cdot U_{\ell _i}} = \gamma _i \Gamma _{\ell _i} \gamma _i^{-1}$ and $H^{\prime }_{i} = \Gamma ^{\prime }_{h(\gamma _i \cdot U_{\ell _i})}$ for $i> 0$ . Then, $H_{j} \subset H_i \subset H = \Gamma _U$ and $H^{\prime }_{j} \subset H^{\prime }_i \subset H' = \Gamma ^{\prime }_{U'}$ for $j> i > 0$ .

Let $\{X_{\infty } : U\}$ denote the set of translates of the adapted set U. Then, for $H = \Gamma _U$ , we have the set equality $\{X_{\infty } : U\} = \Gamma /H$ . Set $m = \#(\Gamma /H)!$ , which is the order of the group of permutations on the set $\{X_{\infty } : U\}$ . Then, for $\gamma \in \Gamma $ , the action of $\gamma ^m$ on the set $\{X_{\infty } : U\}$ is the identity. This implies that $\gamma ^m \in C^{\Gamma }_H$ the core of H.

Given a group chain ${\mathcal G}$ and $H \subset \Gamma $ a subgroup with $\Gamma _{\ell } \subset H$ for some $\ell \geq 1$ , the H-restricted order of $\gamma \in \Gamma $ is the Steinitz order with respect to the chain ${\mathcal C}_{\gamma }^H$ ,

(4.14) $$ \begin{align} \xi^H(\gamma) = \mathrm{lcm} \{ \#( \langle \gamma \rangle /C_{\gamma, \ell}^H ) \mid \ell> 0 \}. \end{align} $$

The H-restricted typeset for the chain ${\mathcal G}$ is the collection

(4.15) $$ \begin{align} \Xi_H[{\mathcal G}] = \{ \tau[\xi^H(\gamma)] \mid \gamma \in \Gamma\}. \end{align} $$

So without loss of generality, it suffices to consider $\gamma \in C^{\Gamma }_H \subset H$ .

The homeomorphism $h \colon U \to U'$ induces an isomorphism $h_* \colon {\mathcal H}_U \to {\mathcal H}^{\prime }_{U'}$ . Thus, there exists $\gamma ' \in H'$ whose action $\Phi ^{\prime }_{U'}(\gamma ')$ on $U'$ equals the image $\phi ^{\prime }_{\gamma } = h_*(\Phi _U(\gamma ))$ . We show that the $H'$ -restricted type of $\gamma '$ equals the H-restricted type of $\gamma $ .

Let $V \subset W \subset X_{\infty }$ be adapted subsets. The restricted action of $\Gamma _W$ on W is minimal, and hence the translates of V define a clopen partition of W. Let $\{W:V\}$ denote the set of elements in this partition and let $|W:V| = \#\{W:V\}$ denote the cardinality of the set of translates.

The action $\Phi $ induces a map $\Phi _{V}^W \colon \Gamma _W \to \mathrm {Aut}(\{W:V\})$ into the permutations of the set $\{W:V\}$ of translates of V in W. The kernel of the map $\Phi _{V}^W$ is denoted by $C_V^W \subset \Gamma _V$ and equals the normal core of $\Gamma _V$ as a subgroup of $\Gamma _W$ . Thus, the index $[\Gamma _W: \Gamma _V] = |W:V|$ and $[\Gamma _W: C^W_V] = \# \mathrm {Image}(\Phi _{V}^W(\Gamma _W))$ .

Now, apply this observation to the action of $H = \Gamma _U$ on U. For each $j \geq 1$ , the action $\Phi $ induces a map $\widehat {\Phi }_{U_{\ell _j}}^U \colon H \to \mathrm {Aut}(\{U : \gamma _j \cdot U_{\ell _j}\})$ , which permutes the elements of this partition. The kernel of the action map $\widehat {\Phi }_{U_{\ell _j}}^U$ is the subgroup

(4.16) $$ \begin{align} C^H_j = \bigcap_{\delta \in H} ~ \delta \Gamma_{\gamma_j \cdot U_{\ell_j}} \delta^{-1} = \bigcap_{\delta \in H} ~( \delta \gamma_j) \Gamma_{\ell_j} (\delta \gamma_j )^{-1} = \bigcap_{\delta \in H} ~ \delta \Gamma_{\ell_j} \delta^{-1}. \end{align} $$

Set $C_{\gamma , j}^H = \langle \gamma \rangle \cap C_{j}^H$ , then the subgroup $\langle \gamma \rangle /C_{\gamma , j}^H \subset H/C_{j}^H$ is mapped injectively into $\mathrm {Aut}(\{U : U_{\ell _j}\})$ . Thus, $\#( \langle \gamma \rangle /C_{\gamma , j}^H ) = \# \mathrm {Image}(\Phi _{U_{\ell _j}}^U(\langle \gamma \rangle ))$ , and so we have

(4.17) $$ \begin{align} \xi^H(\gamma) = \mathrm{lcm} \{ \#( \langle \gamma \rangle /C_{\gamma, j}^H ) \mid j> 0 \} = \mathrm{lcm} \{ \# \mathrm{Image}(\widehat{\Phi}_{U_{\ell_j}}^U(\langle \gamma \rangle)) \mid j > 0 \}. \end{align} $$

We next use the conjugation by h to relate the order of a subgroup of $\mathrm {Aut}(\{U : V\})$ with the order of a subgroup of $\mathrm {Aut}(\{U' : V'\})$ for appropriate choices of adapted subsets V and $V'$ . This is analogous to the idea behind the proof of Lemma 4.1.

From diagram (4.1), for $j \geq 1$ , we have

(4.18) $$ \begin{align} U' \supset h(\gamma_1 \cdot U_{\ell_1}) \supset \gamma^{\prime}_1 \cdot U^{\prime}_{\ell^{\prime}_1} \supset \cdots \supset h(\gamma_{j} \cdot U_{\ell_j}) \supset \gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j} \supset \cdots. \end{align} $$

Using that $\gamma ^{\prime }_j \cdot U^{\prime }_{\ell ^{\prime }_j} \subset h(\gamma _{j} \cdot U_{\ell _j})$ , we have

(4.19) $$ \begin{align} \gamma^{\prime}_{j} \Gamma^{\prime}_{\ell_{j}}(\gamma^{\prime}_{j})^{-1} = \Gamma^{\prime}_{\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j}} \subset \Gamma^{\prime}_{h(U_{\ell_j})} = H^{\prime}_j \subset H' , \end{align} $$

and so $C^{H'}_j = C^{H'}_{\gamma ^{\prime }_{j} \Gamma ^{\prime }_{\ell _{j}}(\gamma ^{\prime }_{j})^{-1}} \subset C^{H'}_{H^{\prime }_j}$ . Then,

(4.20) $$ \begin{align} \langle \gamma' \rangle \supset ( \langle \gamma' \rangle \cap C^{H'}_{H^{\prime}_j} ) \supset ( \langle \gamma' \rangle \cap C^{H'}_{j} ) = C^{H'}_{\gamma',j}. \end{align} $$

The action of $\Phi ^{\prime }_{U'}(\gamma ')$ on $U'$ translates the clopen set $h(\gamma _j \cdot U_{\ell _j})$ within the clopen set $U'$ , and thus

(4.21) $$ \begin{align} \#( \langle \gamma \rangle /C_{\gamma, j}^H ) = \# \mathrm{Image}(\widehat{\Phi}_{U_{\ell_j}}^U(\langle \gamma \rangle)) = \# \mathrm{Image}(\widehat{\Phi}_{h(U_{\ell_j})}^{U'}(\langle \gamma' \rangle)) = \#( \langle \gamma' \rangle /(\langle \gamma' \rangle \cap C^{H'}_{H^{\prime}_j} ) ). \end{align} $$

Using that $\gamma ^{\prime }_j \cdot U^{\prime }_{\ell ^{\prime }_j} \subset h(\gamma _{j} \cdot U_{\ell _j})$ , we have

(4.22) $$ \begin{align} \gamma^{\prime}_{j} \Gamma^{\prime}_{\ell_{j}}(\gamma^{\prime}_{j})^{-1} = \Gamma^{\prime}_{\gamma^{\prime}_j \cdot U^{\prime}_{\ell^{\prime}_j}} \subset \Gamma^{\prime}_{h(U_{\ell_j})} = H^{\prime}_j \subset H' , \end{align} $$

and so $C^{H'}_j = C^{H'}_{\gamma ^{\prime }_{j} \Gamma ^{\prime }_{\ell _{j}}(\gamma ^{\prime }_{j})^{-1}} \subset C^{H'}_{H^{\prime }_j}$ . We then have, using equation (4.20),

(4.23) $$ \begin{align} \#( \langle \gamma' \rangle /C_{\gamma', j}^{H'} ) & = \#( \langle \gamma' \rangle /(\langle \gamma' \rangle \cap C^{H'}_{H^{\prime}_j} ) ) \cdot \#((\langle \gamma' \rangle \cap C^{H'}_{H^{\prime}_j} ) / C_{\gamma', j}^{H'} ) \nonumber\\ & = \#( \langle \gamma \rangle /C_{\gamma, j}^H ) \cdot \#((\langle \gamma' \rangle \cap C^{H'}_{H^{\prime}_j} ) / (\langle \gamma' \rangle \cap C^{H'}_{j} ) ). \end{align} $$

Thus, we have that $\#( \langle \gamma \rangle /C_{\gamma , j}^H )$ divides $\#( \langle \gamma ' \rangle /C_{\gamma ', j}^{H'} ) $ for all $j \geq 1$ . It follows that $\tau _H[\gamma ] = \tau [\xi ^{H}(\gamma )] \leq \tau [\xi ^{H'}(\gamma ')] = \tau _{H'}[\gamma ']$ . Then, by reversing these calculations, starting with $\gamma '$ chosen as above, we obtain $\tau _{H'}[\gamma '] \leq \tau _H[\gamma ]$ and hence $\tau _H[\gamma ] = \tau _{H'}[\gamma ']$ .

We have thus shown that for each $\gamma \in \Gamma $ , there exists $\gamma ' \in \Gamma '$ with the same restricted type. Reversing this process, we deduce that $\Xi _H[X_{\infty },\Gamma ,\Phi ] = \Xi _{H'}[X^{\prime }_{\infty },\Gamma ',\Phi ']$ . It follows that $\Xi [{\mathcal G}]$ and $\Xi [{\mathcal G}']$ are commensurable as in Definition 3.9, which proves Theorem 1.8.

4.3 Proof of Theorem 1.7

The claim of Theorem 1.7 is that for each $\gamma \in \Gamma $ , there is a well-defined type $\tau [\gamma ]$ . In fact, we show the stronger result, that the Steinitz order $\xi (\gamma )$ is independent of the choice of an adapted basis ${\mathcal U}$ for the odometer $({\mathfrak {X}}, \Gamma , \Phi )$ . Given two adapted bases ${\mathcal U}$ and ${\mathcal U}'$ , the algebraic models they define, $(X_{\infty }, \Gamma , \Phi _{\infty })$ and $(X^{\prime }_{\infty }, \Gamma , \Phi _{\infty }')$ , are isomorphic.

In the above proof of Theorem 1.8, we let $U = X_{\infty }$ and $U' = X^{\prime }_{\infty }$ , and so $H = \Gamma _U = \Gamma $ and $H' = \Gamma ^{\prime }_{U'} = \Gamma '$ . Then, observe that for $\ell> 0$ , the normal subgroups $C^H_{\ell } = C_{\ell }$ and $C^{H'}_{\ell } = C^{\prime }_{\ell }$ . It follows that for $\gamma \in \Gamma $ and $\gamma ' \in \Gamma '$ chosen as in the proof of Theorem 1.8, we have $\xi ^H(\gamma ) = \xi (\gamma )$ and $\xi ^{H'}(\gamma ') = \xi (\gamma ')$ , and the proof shows that $\xi (\gamma ) = \xi (\gamma ')$ . It follows that the set of orders $\{ \xi (\gamma ) \mid \gamma \in \Gamma \}$ is independent of the choice of an algebraic model for the action, and so is an invariant of the isomorphism class of the action $({\mathfrak {X}}, \Gamma , \Phi )$ , as was to be shown.

4.4 Proof of Corollary 1.9

Suppose that both $\Gamma $ and $\Gamma '$ are abelian, and the actions are return equivalent. Then, every group $\Gamma _{\ell }$ in a subgroup chain ${\mathcal G}_{{\mathcal U}}$ in $\Gamma $ is normal. Then, for any subgroup $H \subset \Gamma $ , the normal cores satisfy $C^H_{\ell } = C_{\ell } = \Gamma _{\ell }$ . Thus, in equation (4.17), the restricted type $\xi ^H(\gamma ) = \xi (\gamma )$ . Similarly, we have $\xi ^{H'}(\gamma ') = \xi (\gamma ')$ for $\gamma ' \in \Gamma '$ . Thus, the above proof shows in this case that the typesets $\Xi [{\mathfrak {X}},\Gamma ,\Phi ]$ and $\Xi [{\mathfrak {X}}',\Gamma ',\Phi ']$ are equal, which is the claim of Corollary 1.9.

It is surprising perhaps that the conclusion of Corollary 1.9 need not hold if one of the groups is virtually abelian, but not abelian, as illustrated in Example 5.3. The idea of Example 5.3 is that we add to an abelian group a single element that normalizes it, does not commute with it, and destroys the equality $C^H_{\ell } = C_{\ell }$ . Then, the types of elements are no longer equal to their H-restricted types.

4.5 Proof of Theorem 1.10

Let $\varphi \colon \Gamma \to \Gamma $ be a self-embedding with finite index image. Define a group chain ${\mathcal G}_{\varphi }$ by setting $\Gamma _0 = \Gamma $ and then inductively define $\Gamma _{\ell +1} = \varphi (\Gamma _{\ell }) \subset \Gamma _{\ell }$ for $\ell \geq 0$ .

Then, with $H = \Gamma _{k}$ for $k> 0$ and $\ell> k$ ,

(4.24) $$ \begin{align} C^H_{\ell} = \bigcap_{\delta \in H} \ \delta^{-1} \Gamma_{\ell} \delta = \bigcap_{\delta \in \varphi^{k}(\Gamma)} \ \delta^{-1} \varphi^{\ell}(\Gamma) \delta = \varphi^k\bigg( \bigcap_{\delta \in \Gamma} \ \delta^{-1} \varphi^{\ell-k}(\Gamma) \delta \bigg) = \varphi^k(C_{\ell-k}) , \end{align} $$

and so $\Gamma _{\ell }/C^H_{\ell } = \varphi ^k(\Gamma _{\ell -k}/C_{\ell -k})$ . Then, for $\gamma \in H = \Gamma _{k}$ , set $\overline {\gamma } = \varphi ^{-k}(\gamma )$ , then we have

(4.25) $$ \begin{align} \!\!\!\!\!\xi^H(\gamma) = \mathrm{lcm} \{ \#( \langle \gamma \rangle /C_{\gamma, \ell}^H ) \mid \ell> k \} = \mathrm{lcm} \{ \#( \langle \overline{\gamma} \rangle /C_{\overline{\gamma}, \ell-k} ) \mid \ell -k > 0 \} = \xi(\overline{\gamma}). \end{align} $$

Then, as $\Gamma = \varphi ^{-k}(H)$ , we have $\Xi _H[X_{\varphi }, \Gamma , \Phi _{\varphi }] = \Xi [X_{\varphi }, \Gamma , \Phi _{\varphi }]$ .

Let $(X_{\varphi }, \Gamma , \Phi _{\varphi })$ and $(X_{\varphi '}, \Gamma ', \Phi _{\varphi '})$ be odometers associated to renormalizations $\varphi \colon \Gamma \to \Gamma $ and $\varphi ' \colon \Gamma ' \to \Gamma '$ . Assume the actions are return equivalent by a homeomorphism $h \colon U \to U'$ . Then, choose $\delta \in \Gamma $ with $\delta \cdot e_{\infty } \in U$ and $\delta ' \in \Gamma '$ with $\delta ' \cdot e^{\prime }_{\infty } \in U'$ . Then, $\varphi ^{\delta } = \Phi (\delta ) \circ \varphi \circ \Phi (\delta ^{-1})$ is a renormalization of $\Gamma $ with group chain $\Gamma ^{\delta }_{\ell } = \delta \Gamma _{\ell } \delta ^{-1}$ . The group $\Gamma ^{\delta }_{\ell }$ stabilizes the translate $\delta \cdot U_{\ell }$ . As $\delta \cdot e_{\infty } \in U$ , there exists $k> 0$ such that $\delta \cdot U_k \subset U$ . Repeat this argument for the renormalization $\varphi '$ to obtain a $\delta ' \in \Gamma '$ and $k '> 0$ such that $\delta ' \cdot U^{\prime }_{k'} \subset U'$ .

Then, proceed as in the proof in §4.2 with $H = \delta \Gamma _{k} \delta ^{-1}$ and $H' = \delta ' \Gamma ^{\prime }_{k'} (\delta ')^{-1}$ to obtain that

$$ \begin{align*}\Xi[X_{\varphi}, \Gamma, \Phi_{\varphi}] = \Xi_H[X_{\varphi}, \Gamma, \Phi_{\varphi}] = \Xi_{H'}[X^{\prime}_{\varphi'}, \Gamma', \Phi_{\varphi'}] = \Xi[X^{\prime}_{\varphi'}, \Gamma', \Phi_{\varphi'}],\end{align*} $$

as claimed in Theorem 1.10. In particular, this implies that the typeset $\Xi [X_{\varphi }, \Gamma , \Phi _{\varphi }]$ is an invariant of the isomorphism class of the renormalizable Cantor action $(X_{\varphi }, \Gamma , \Phi _{\varphi })$ .

5.1 Virtually abelian actions

A group $\Gamma $ is virtually abelian if it admits a finite-index subgroup $A \subset \Gamma $ which is abelian. It is straightforward to construct ${\mathbb Z}^n$ -odometer actions with prescribed spectra, and yet they illustrate several basic properties of the type and typeset invariants.

Example 5.1. Consider the case $n=1$ . Choose two disjoint sets of distinct primes,

$$ \begin{align*}\pi_f = \{q_1 , q_2, \ldots \}, \quad \pi_{\infty} = \{p_1 , p_2, \ldots\},\end{align*} $$

where $\pi _f$ and $\pi _{\infty }$ can be chosen to be finite or infinite sets, and either $\pi _f$ is infinite or $\pi _{\infty }$ is non-empty. Choose multiplicities $n(q_i) \geq 1$ for the primes in $\pi _f$ . For each $\ell> 0$ , define a subgroup of $\Gamma = {\mathbb Z}$ by

(5.1) $$ \begin{align} \Gamma_{\ell} = \{q_1^{n(q_1)} q_2^{n(q_2)} \cdots q_{\ell}^{n(q_{\ell})} \cdot p_1^{\ell} p_2^{\ell} \cdots p_{\ell}^{\ell} \cdot n \mid n \in {\mathbb Z} \}. \end{align} $$

If $\pi _{\infty }$ is a finite set, then we use the convention that $p_{\ell } =1$ in equation (5.1) when $p_{\ell }$ is not defined by the listing of $\pi _{\infty }$ . The completion $\widehat {\Gamma }$ of ${\mathbb Z}$ with respect to this group chain admits a product decomposition into its Sylow p-subgroups

(5.2) $$ \begin{align} \widehat{\Gamma} ~ \cong ~ \prod_{i =1}^{\infty} \ {\mathbb Z}/q_i^{n(q_i)} {\mathbb Z} ~ \cdot ~ \prod_{p \in \pi_{\infty}} ~ \widehat{\mathbb Z}_{(p)} , \end{align} $$

where $\widehat {\mathbb Z}_{(p)}$ denotes the p-adic completion of ${\mathbb Z}$ . Thus, $\pi (\xi (\widehat {\Gamma })) = \pi _f \cup \pi _{\infty }$ . As ${\mathbb Z}$ is abelian, $X_{\infty } = \widehat {\Gamma }$ . The type of this action classifies it up to return equivalence.

Example 5.3. We next give an example that illustrates that the commensurable relation on typesets is optimal. We construct the simplest example which shows this, and it is clear that many more similar constructions are possible.

Let $\Gamma = {\mathbb Z}^2 \rtimes {\mathbb Z}_2$ be the semi-direct product of ${\mathbb Z}^2$ with the order $2$ group ${\mathbb Z}_2 = {\mathbb Z}/2{\mathbb Z}$ , where the generator $\sigma \in {\mathbb Z}_2$ acts on ${\mathbb Z}^2$ by permuting the summands. Let $\Gamma ' = {\mathbb Z}^2$ , which is abelian.

Choose distinct primes $p, q> 1$ . Define the subgroup chain in $\Gamma $ and $\Gamma '$ as follows:

(5.3) $$ \begin{align} \Gamma_{\ell} = \{(p^{\ell} k, q^{\ell} m, id) \mid (h,m) \in {\mathbb Z}^2\},\quad \Gamma^{\prime}_{\ell} = \{(p^{\ell} k, q^{\ell} m) \mid (k,m) \in {\mathbb Z}^2\}. \end{align} $$

Note that the resulting actions $(X_{\infty }, \Gamma , \Phi )$ and $(X^{\prime }_{\infty }, \Gamma ', \Phi ')$ are return equivalent.

Observe that $C^{\prime }_{\ell } = \Gamma ^{\prime }_{\ell }$ . However, $\Gamma _{\ell }$ is not normal in $\Gamma $ and we have

(5.4) $$ \begin{align} C_{\ell}=\{((pq)^{\ell} k, (pq)^{\ell} m, \mathrm{id}) \mid (h,m) \in {\mathbb Z}^2\} \subset \Gamma_{\ell}. \end{align} $$

The actions of $\Gamma $ and $\Gamma '$ have the same type, with characteristic functions ${\chi (p) = \chi (q) = \infty }$ , and all other values are zero. However, we have the typesets

(5.5) $$ \begin{align} \Xi[X_{\infty}, \Gamma, \Phi] = \{ [(pq)^{\infty}] \} \quad \mathrm{and} \quad \Xi[X^{\prime}_{\infty}, \Gamma', \Phi'] = \{ [p^{\infty}] , [q^{\infty}], [(pq)^{\infty}]\} , \end{align} $$

so the typeset is not invariant under return equivalence.

This example easily generalizes, where we take $\Gamma ' = {\mathbb Z} \oplus \cdots \oplus {\mathbb Z}$ to be the direct sum of n copies of ${\mathbb Z}$ , and replace ${\mathbb Z}_2$ with any non-trivial subgroup $\Delta \subset \mathrm {Perm}(n)$ of the permutation group on n elements. Let $\Gamma = \Gamma ' \rtimes \Delta $ be the semi-direct product of $\Gamma '$ with $\Delta $ . Choose the subgroup chain $\{\Gamma ^{\prime }_{\ell '}\}$ in $\Gamma '$ as in Example 5.2 and use the chain $\{\Gamma _{\ell } = \Gamma ^{\prime }_{\ell } \times \mathrm {id} \}$ in $\Gamma $ . Then, the resulting actions are return equivalent and one can obtain a wide variety of finite typesets $\Xi [X^{\prime }_{\infty }, \Gamma ', \Phi ']$ for the abelian action. If the action of $\Delta $ is transitive, then we have that $\Xi [X_{\infty }, \Gamma , \Phi ]$ consists of a single element. When $\Delta $ does not act transitively, there is even more variation on the typesets of the two actions.

5.2 Nilpotent odometers

The next examples use the actions associated to a renormalization of a finitely generated group $\Gamma $ . Many finitely generated nilpotent groups admit a renormalization [Reference Cornulier12, Reference Dekimpe and Deré15, Reference Dekimpe and Lee16, Reference Endimioni and Robinson19, Reference Lee and Lee30, Reference Nekrashevych and Pete36], which yield many examples of odometer actions with well-defined typesets by Theorem 1.10.

The integer Heisenberg group is the simplest non-abelian nilpotent group, and is represented as the upper triangular matrices in $\mathrm {GL}(3,{\mathbb Z})$ . That is,

(5.6) $$ \begin{align} \Gamma = \left\{ \left[ {\begin{array}{ccc} 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end{array} } \right] \mid a,b,c \in {\mathbb Z}\right\}. \end{align} $$

We denote a $3 \times 3$ matrix in $\Gamma $ by the coordinates as $(a,b,c)$ .

Example 5.4. For a prime $p \geq 2$ , define the self-embedding $\varphi _p \colon \Gamma \to \Gamma $ by $\varphi (a,b,c) = (pa, pb, p^2c)$ . Then, define a group chain in $\Gamma $ by setting

$$ \begin{align*}\Gamma_{\ell} = \varphi_p^{\ell}(\Gamma) = \{(p^{\ell} a, p^{\ell}b, p^{2\ell}c) \mid a,b,c \in {\mathbb Z}\}, \quad \bigcap_{\ell> 0} \ \Gamma_{\ell} = \{e\} .\end{align*} $$

For $\ell> 0$ , the normal core for $\Gamma _{\ell }$ is given by $C_{\ell } = \mathrm {core}(\Gamma _{\ell }) = \{(p^{2\ell } a, p^{2\ell } b, p^{2\ell } c) \mid a,b,c \in {\mathbb Z}\}$ , and so the quotient group $Q_{\ell } = \Gamma /C_{\ell } \cong \{( \overline {a}, \overline {b}, \overline {c}) \mid \overline {a}, \overline {b}, \overline {c} \in {\mathbb Z}/p^{2\ell }{\mathbb Z} \}$ . The profinite group $\widehat {\Gamma }_{\infty }$ is the inverse limit of the quotient groups $Q_{\ell }$ , so we have $ {\widehat {\Gamma }_{\infty } = \{(\widehat {a}, \widehat {b}, \widehat {c}) \mid \widehat {a}, \widehat {b}, \widehat {c}\in \widehat {{\mathbb Z}}_{p^2} \}}$ . Thus, every non-trivial $\gamma \in \Gamma $ has type $\tau [\gamma ] = \tau [p^{\infty }]$ .

Example 5.5. For distinct primes $p, q \geq 2$ , define the self-embedding $\varphi _{p,q} \colon \Gamma \to \Gamma $ by $\varphi (a,b,c) = (pa, qb, pqc)$ . Then, define a group chain in $\Gamma $ by setting

$$ \begin{align*}\Gamma_{\ell} = \varphi_{p,q}^{\ell}(\Gamma) = \{(p^{\ell} a, q^{\ell}b, (pq)^{\ell}c) \mid a,b,c \in {\mathbb Z}\} ,\quad \bigcap_{\ell> 0} \ \Gamma_{\ell} = \{e\} .\end{align*} $$

For $\ell> 0$ , the normal core for $\Gamma _{\ell }$ is given by $C_{\ell } = \mathrm {core}(\Gamma _{\ell }) = \{((pq)^{\ell } a, (pq)^{\ell } b, (pq)^{\ell } c) \mid a,b,c \in {\mathbb Z}\}$ , and so we obtain the quotient group $Q_{\ell } = \Gamma /C_{\ell } \cong \{( \overline {a}, \overline {b}, \overline {c}) \mid \overline {a}, \overline {b}, \overline {c} \in {\mathbb Z}/(pq)^{\ell }{\mathbb Z} \}$ . The profinite group $\widehat {\Gamma }_{\infty }$ is the inverse limit of the quotient groups $Q_{\ell }$ , so we have $ \widehat {\Gamma }_{\infty } = \{(\widehat {a}, \widehat {b}, \widehat {c}) \mid \widehat {a}, \widehat {b}, \widehat {c}\in \widehat {{\mathbb Z}}_{pq} \}$ . Thus, every non-trivial $\gamma \in \widehat {\Gamma }_{\infty }$ has type $\tau [\gamma ] = \tau [(pq)^{\infty }]$ .

Note that the typeset for the odometer defined by the $\varphi _{p,q}$ -renormalization equals the typeset for the abelian action in Example 5.3, but the two actions are clearly not return equivalent.

A second source of examples for odometer actions of nilpotent groups uses the decomposition of a profinite nilpotent group into its prime localizations, a technique that is especially adapted to realizing a given collection of primes as the spectrum of such an action. Moreover, these actions can be constructed to have special dynamical properties, as in [Reference Hurder and Lukina27]. The construction is necessarily more complex than for renormalizable actions, as there must be an infinite sequence of choices to make. These ideas were developed in [Reference Hurder and Lukina24, §9] for actions of $SL(n, {\mathbb Z})$ , and the work [Reference Hurder and Lukina28] discusses this construction for nilpotent groups.

6.1 Actions on trees

A tree T is an infinite graph with the set of vertices $V = \bigsqcup _{\ell \geq 0} V_\ell $ and the set of edges E. Each $V_\ell $ , $\ell \geq 0$ is a finite set, called the set of vertices at level $\ell $ . Edges in E join pairs of vertices in consecutive level sets $V_{\ell +1}$ and $V_\ell $ , $\ell \geq 0$ , so that a vertex in $V_{\ell +1}$ is connected to a single vertex in $V_\ell $ by a single edge. A tree is rooted if $|V_0| = 1$ .

We assume that $n_\ell \geq 2$ for $\ell \geq 1$ . If $d = 2$ , then a $2$ -ary tree T is also called a binary tree.

Let $({\mathfrak {X}},\Gamma ,\Phi )$ be an odometer and let ${\mathcal U} = \{U_{\ell } \subset {\mathfrak {X}} \mid \ell> 0\}$ be a choice of an adapted neighborhood basis. By Corollary 2.4, there is a group chain $ {\mathcal G}_{{\mathcal U}} = \{\Gamma _{\ell } = \Gamma _{U_{\ell }} \mid \ell \geq 0\}$ such that, associated to ${\mathcal G}_{{\mathcal U}}$ is an odometer $(X_{\infty },\Gamma ,\Phi _{\infty })$ which is isomorphic to $({\mathfrak {X}},\Gamma ,\Phi )$ . Here, $X_{\infty }$ is the inverse limit space of finite sets $X_\ell = \Gamma /\Gamma _\ell $ , given by equation (1.3), see §2.3 for details.

A tree model for the action $({\mathfrak {X}},\Gamma ,\Phi )$ is constructed using the group chain ${\mathcal G}_{{\mathcal U}}$ . For $\ell \geq 0$ , let $V_\ell = X_\ell $ , and join $v_\ell \in V_\ell $ and $v_{\ell +1} \in V_{\ell +1}$ by an edge if and only if $v_{\ell +1} \subset v_\ell $ as cosets. The tree so constructed is spherically homogeneous, with spherical index entries $n_\ell = |\Gamma _{\ell -1}: \Gamma _{\ell }|$ for $\ell \geq 1$ . The boundary $\partial T$ of T is the collection of all infinite paths in T, that is,

$$ \begin{align*}\partial T = \bigg\{(v_\ell)_{\ell \geq 0} \subset \prod_{\ell \geq 0} V_\ell \mid v_{\ell+1} \textrm{ and }v_\ell \textrm{ are joined by an edge}\bigg\} \cong X_{\infty},\end{align*} $$

and the induced action of $\Gamma $ on $\partial T$ , which we also denote by $(X_{\infty },\Gamma ,\Phi )$ .

6.2 Typesets of regular actions

We consider next the typeset of a d-regular odometer.

Theorem 6.4. An d-regular odometer $({\mathfrak {X}},\Gamma ,\Phi )$ has finite typeset $\Xi [\mathfrak {X},\Gamma ,\Phi ]$ . More precisely, suppose $({\mathfrak {X}},\Gamma ,\Phi )$ is isomorphic to an action on a d-ary rooted tree for some $d \geq 2$ . Let $P_d$ be the set of distinct prime divisors of the elements in the collection $\{2,3,\ldots ,d\}$ and let $N_d = |P_d|$ . Then,

(6.1) $$ \begin{align} |\Xi[{\mathfrak{X}},\Gamma,\Phi]| \leq \sum_{k=0}^{N_d} \binom{N_d}{k} = \sum_{k=0}^{N_d} \frac{N_d!}{(N_d - k)! k!}. \end{align} $$

Moreover, each equivalence class $\tau \in \Xi [{\mathfrak {X}},\Gamma ,\Phi ]$ is represented by a Steinitz number $\xi $ with empty finite prime spectrum, $\pi _f(\xi ) = \emptyset $ , and so $\pi (\xi ) = \pi _{\infty }(\xi )$ .

Example 6.7. Let p be an odd prime. The Gupta–Sidki p-group $GS(p)$ [Reference Nekrashevych35, §1.8.1] is a group acting on the rooted p-ary tree. The group is generated by a cyclic permutation of the set of p elements $\sigma = (0,1,\ldots ,p-1)$ and the recursively defined map

$$ \begin{align*}\tau = (\sigma, \sigma^{-1},1, \ldots, \tau).\end{align*} $$

Every element in the Gupta–Sidki p-group has finite order and so $\Xi [X_{\infty },GS(p), \Phi _{\infty }] = \{[1]\}$ , where $[1]$ is the trivial type, that is, the type of the identity element.

6.3 Typeset under the commensuration relation

We now show that H-restricted types of a group element and of its conjugate need not coincide.

Given a tree T and a vertex $v \in V$ , denote by $T_v$ the subtree of T with root v.

7.1 Type invariants for solenoidal manifolds

The covering degree $m_{\ell }$ of $q_{\ell } \colon M_{\ell } \to M_{\ell -1}$ equals the index of the subgroup $[\Gamma _{\ell -1} : \Gamma _{\ell }]$ , and so the covering degree of $\overline {q}_{\ell } \colon M_{\ell } \to M_0$ is given by

(7.3) $$ \begin{align} \mathrm{deg}(\overline{q}_{\ell}) = m_{\ell} \cdot m_{\ell -1} \cdots m_1 = [\Gamma : \Gamma_{\ell}]. \end{align} $$

The Steinitz order of a presentation $\mathcal P$ is

(7.4) $$ \begin{align} \xi({\mathcal P}) = \mathrm{lcm} \{ m_1 m_2 \cdots m_{\ell} \mid \ell> 0\} = \mathrm{lcm} \{ [\Gamma : \Gamma_{\ell}] \mid \ell > 0\} = \xi({\mathcal G}_{\mathcal P}). \end{align} $$

The Steinitz number $\xi ({\mathcal P}')$ can be thought of as the covering degree of the fibration ${\widehat {q}}_0 \colon {\mathcal M}_{{\mathcal P}} \to M_0$ .

Given a second presentation ${\mathcal P}'$ with solenoidal manifold ${\mathcal M}^{\prime }_{{\mathcal P}'}$ and choice of basepoint $x_0' \in {\mathcal M}^{\prime }_{{\mathcal P}'}$ , we similarly obtain subgroups $\Gamma ^{\prime }_{\ell }$ defining the group chain ${\mathcal G}^{\prime }_{{\mathcal U}'}$ , and we have $\xi ({\mathcal P}') = \xi ({\mathcal G}^{\prime }_{{\mathcal U}'})$ .

Suppose that the solenoidal manifolds ${\mathcal M}_{{\mathcal P}}$ and ${\mathcal M}^{\prime }_{{\mathcal P}'}$ are homeomorphic, then by Theorem 7.1, the corresponding odometers $(X_{\infty },\Gamma ,\Phi _{\infty })$ and $(X_{\infty }',\Gamma ',\Phi _{\infty }')$ are return equivalent, and hence they have the equal types $\tau [X_{\infty },\Gamma ,\Phi _{\infty }] = \tau [X_{\infty }',\Gamma ',\Phi _{\infty }']$ by Theorem 1.5. We thus obtain the following theorem.

We thus obtain the following well-defined homeomorphism invariants for solenoidal manifolds.

For orientable one-dimensional solenoids, where each map $q_{\ell } \colon {\mathbb S}^1 \to {\mathbb S}^1$ is orientable, Bing observed in [Reference Bing7] that if $\tau [{\mathcal M}_{{\mathcal P}}] = \tau [{\mathcal M}_{{\mathcal P}'}]$ , then ${\mathcal M}_{\mathcal P}$ and ${\mathcal M}_{{\mathcal P}'}$ are homeomorphic. McCord showed in [Reference McCord32, §2] the converse, that if ${\mathcal M}_{\mathcal P}$ and ${\mathcal M}_{{\mathcal P}'}$ are homeomorphic, then $\tau [{\mathcal M}_{{\mathcal P}}] = \tau [{\mathcal M}_{{\mathcal P}'}]$ . (See also [Reference Aarts and Fokkink1].) Together, these results yield the following well-known classification.

For solenoidal manifolds of dimension $n \geq 2$ , no such classification by invariants can exist, even when the base manifold $M_0 = {\mathbb T}^n$ . The most one can hope for is to define invariants which distinguish various homeomorphism classes, such as the type and typeset for a solenoidal manifold, invariants derived from the Cantor action associated to the monodromy of the fibration map ${\widehat {q}}_0 \colon {\mathcal M}_{{\mathcal P}} \to M_0$ .

Note that by Theorem 7.2, if two toroidal solenoids have return equivalent monodromy odometers, then they are homeomorphic. Thus, in this special case, the classification problem for solenoidal manifolds is equivalent to the classification problem of ${\mathbb Z}^n$ -odometers modulo return equivalence. Giordano, Putnam, and Skau discuss in [Reference Giordano, Putnam and Skau22] this classification problem.

7.2 Typeset invariants for solenoidal manifolds

The typeset invariants for solenoidal manifolds provide more refined invariants of their homeomorphism class. Before stating the precise definition, we give an intuitive definition.

Let $M_0$ be the base manifold of a presentation ${\mathcal P}$ . Let $x_{\infty } \in {\mathcal M}_{{\mathcal P}}$ be a choice of a basepoint, which determines the basepoint $x_0 \in M_0$ and fundamental group ${\Gamma = \pi _1(M_0, x_0)}$ . Suppose that $\gamma \in \Gamma $ is represented by a simple closed curve $c_{\gamma } \colon {\mathbb S}^1 \to M_0$ . The projection map ${\widehat {q}}_0 \colon {\mathcal M}_{{\mathcal P}} \to M_0$ restricts to a covering map on each leaf of the foliation ${\mathcal F}_{{\mathcal P}}$ of ${\mathcal M}_{{\mathcal P}}$ . When the preimage ${\mathcal M}_{\gamma } = {\widehat {q}}_0^{-1}(c_{\gamma }({\mathbb S}^1))$ is connected, it is a one-dimensional solenoid and so has a well-defined type by Theorem 7.3. Denote this type by $\tau [\gamma ]$ . The collection of these types is the ‘intuitive typeset’ of ${\mathcal M}_{{\mathcal P}}$ .

The formal definition of the typeset proceeds as in §7.1. Associate to a presentation ${\mathcal P}$ the monodromy odometer $(X_{\infty }, \Gamma , \Phi _{\infty })$ . Then, for each $\gamma \in \Gamma $ , there is the type $\tau [\gamma ]$ as in Definition 1.6.