# A variant on the Higman-Thompson groups

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 $$G$$.)

• 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 $d=0$). – YCor Aug 31 at 7:46
• 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 $D$. – Colin Reid Aug 31 at 9:31