Let $C = \mathbb{Z}/d\mathbb{Z}$ ($d \ge 0$).

Let $D = \langle a_c : c \in C, t \mid a^2_c = t^d = 1, ta_ct^{-1} = a_{c+1} \rangle$.

let $E$ be the subgroup generated by $\{a_c : c \in C\}$ and let $T$ be the Cayley graph of $E$ with this generating set (note that $T$ is a tree). There is then a natural action of $D$ on $T$, where $E$ acts by left translation and $t$ acts by conjugation on $E$.

(edit) Let $P$ be the product of all pairs of complementary half-trees, equipped with the product topology. Then I think (?) for $d \neq 1$ there will be a minimal closed $D$-invariant subspace $X$ of $P$, consisting of the closure of the set of points in $P$ defined by infinite rays in $T$. Finally, define $G$ to be the topological full group of the action of $D$ on $X$.

The space $X$ is homeomorphic to the Cantor set except for $d \in \{1,2\}$. If $d > 2$, I think the description of $G$ I gave is just another way to build the Higman–Thompson group $G_{d-1,d}$, which is very well-studied. In particular, as shown by Higman, it is a finitely presented group that is either simple ($d$ odd) or has a simple subgroup of index $2$ ($d$ even). But the case $d=0$ also seems fairly natural to me.

Has this particular group (where $d=0$) been studied? If so, what basic properties are known? (I know that some things can be deduced from general results about topological full groups, e.g. that the derived group of $G$ is finitely generated and simple; but for example it's not clear to me how to compute the abelianization of $G$.)

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    $\begingroup$ I think this Stone space construction associated to a tree $T$ is known as Roller compactification $R_T$. As a set it equals the union of the vertex set $V$ and the geodesic boundary. When the tree has finite valency $V$ is open in $R_T$, but not in general (as here for $d=0$). $\endgroup$ – YCor Aug 31 at 7:46
  • $\begingroup$ Ah right, it sounds like the space I described doesn't quite do what I wanted. The Roller compactification is a good place to start, but then I want to get to a minimal closed invariant subspace of that space for the action of $D$. $\endgroup$ – Colin Reid Aug 31 at 9:31

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