The paradigm examples of Polish groups that have finite covering dimension are obviously $\mathbb{R}^n$ for any finite $n$. They are also locally compact. Is it possible to construct a sequence $G_n$ of Polish groups such that
- $G_n$ is not locally compact ($n\in \mathbb{N}$),
- $\dim G_n \to \infty$ as $n\to \infty$?
Note that finite-dimensional Polish spaces are embeddable into Euclidean spaces but not necessarily as groups. Indeed, the group $\mathbb{Z}^{\mathbb{N}}$ is zero-dimensional but it is not isomorphic to a subgroup of an Euclidean space, I think.
Note for downvoters: I agree that the question as stated has trivially negative answer. What I had in mind were groups without non-trivial zero-dimensional direct summands that would be non-locally compact counterparts of familiar Lie groups (in a very broad and vague sense). I did not want to complicate the question, though, which led me to trivialities. Should I apologise?
$G_n = \mathbb{Z}^\mathbb{N} \times \mathbb{R}^n$ will do. A product of separable completely metrisable groups is again one. The dimension is $n$ by standard theorems, and it's not locally compact because $\mathbb{Z}^\mathbb{N}$ is not (no open set can have compact closure; it's homeomorphic as a space to the irrationals, a.k.a Baire space, so it embeds into $\mathbb{R}^{n+1}$, say).