In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in a group $G$ which is the projective limit of infinitely many copies of $F$ (indexed by the natural numbers) with respect to $\alpha$:

$$ G = \{ (a_0, a_1, a_2, \dots) \mid a_i \in F, \alpha(a_{n+1}) = a_n \text{ for all } n \geq 0 \} . $$

Does this construction have a name?

We are applying this specifically in the case where $F$ is the multiplicative group of $\mathbb{C}$ (or any other suitable field) and $\alpha \colon a \mapsto a^p$ for some fixed prime $p$. In this case, our group $G$ contains the group $\mathbb{Q}_p$ of $p$-adic numbers (w.r.t. addition), but it is (much?) larger. Is this group a "known group"?

In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in a group $G$ which is the projective limit of infinitely many copies of $F$ (indexed by the natural numbers) with respect to $\alpha$:

$$ G = \{ (a_0, a_1, a_2, \dots) \mid a_i \in F, \alpha(a_{n+1}) = a_n \text{ for all } n \geq 0 \} . $$

Does this construction have a name?

We are applying this specifically in the case where $F$ is the multiplicative group of $\mathbb{C}$ (or any other suitable field) and $\alpha \colon a \mapsto a^p$ for some fixed prime $p$. In this case, our group $G$ contains the group $\mathbb{Q}_p$ of $p$-adic numbers (w.r.t. addition), but it is (much?) larger. Is this group a "known group"?