What group action underpins the quotient by a group action in this answer?

Question

What group action underpins the quotient by a group action in this answer?

What's the group and what's the operation?

I know what a quotient is. I know what a group action is. I have looked up what a quotient by a group action is, and I'm satisfied I understand the meaning.

But I can't make out what group is acting in the answer.

This is why I'm confused. If it's the action of truncation, that looks like the action of the free monoid on two variables to me and not a group, because e.g. both $\overline01_2$ and $\overline0_2$ truncate to $\overline0_2$ so we don't have a well-defined inverse of truncation. All that seems contradictory to the idea of a group action.

On the other hand, what sense I do seem to be able to make of the answer is that provided we're in $\Bbb Q_2$ and not $\Bbb Z_2$ we might say we can append any $2^nx:x\in\Bbb Z[\frac12]$ to the end of an infinite string to cover $\Bbb Q_2$

Also, a group action necessarily acts on a group. This might be seen as contradictory to the fact I asked for the quotient as a set but I understand a quotient as a group would still satisfy that requirement.

Answer 1

Just for didactic purposes I'll rephrase the correct other answer:

The group which acts here can abstractly be defined as the semidirect product

$$ \mathbb Z[1/p] \rtimes_\varphi \mathbb Z$$

of the two additive groups in the components, with $\varphi(u) \in \mathrm{Aut}(\mathbb Z[1/p] )$ being the map $x \mapsto p^u x$. Concretely, we can write each element of the group as a pair

$$(t,u)$$

with $t\in \mathbb Z[1/p], u \in \mathbb Z$, and the group structure is given by

$$(t,u) \odot (v,w) := (t+p^u v, u+w).$$

A very convenient way to represent this group is as matrices

$$\pmatrix{p^\mathbb Z & \mathbb Z[1/p]\\0&1}$$

i.e. the map $$(t,u) \mapsto \pmatrix{p^u & t\\0&1}$$

gives an emdedding of our group into, say, $\mathrm{GL}_2(\mathbb Q)$. With this, the action alluded to in the other answer (on, say, $x \in \mathbb Q_p$) identifies with the natural action of those matrices on vectors $\pmatrix{x\\1}$.

This is a special case of standard theory of linear fractional transformations.

Answer 2

The group is comprised of functions of the form $x\mapsto p^ux+t$ where $u\in\mathbb{Z}$ and $t\in\mathbb{Z}[1/p]$. You can check this set of functions is a group under composition, and acts on $\mathbb{Q}_p$. We can think of the group as a semidirect product $\mathbb{Z}[1/p]\rtimes\mathbb{Z}$, where conjugating $t\in\mathbb{Z}[1/p]$ by $u\in\mathbb{Z}$ yields $p^ut$.