  Compact finite-rank nilspaces have become central in the nilspace approach to
higher-order Fourier analysis, notably through their role in a general form of
the inverse theorem for the Gowers norms. This paper studies these nilspaces
per se, and in connection with further refinements of this inverse theorem that
have been conjectured recently. Our first main result states that every compact
finite-rank nilspace is obtained by taking a free nilspace (a nilspace based on
an abelian group of the form $\mathbb{Z}^{r}\times \mathbb{R}^s$) and
quotienting this by a discrete group action of a specific type, describable in
terms of polynomials. We call these group actions "higher-order lattice
actions", as they generalize actions of lattices in $\mathbb{Z}^r\times
\mathbb{R}^s$. The second main result (which relies on the first one)
represents every compact finite-rank nilspace as a double-coset space
$K\backslash G / \Gamma$ where $G$ is a nilpotent Lie group of a specific kind.
Our third main result extends the aforementioned results to $k$-step compact
nilspaces (not necessarily of finite rank), by representing any such nilspace
as a quotient of infinite products of free nilspaces and also as double coset
spaces $K\backslash G/\Gamma$ where $G$ is a degree-$k$ nilpotent pro-Lie
group. These results open the study of compact nilspaces to areas more
classical than nilspace theory, such as the theory of topological group
actions. The results also require developing the theory of topological
non-compact nilspaces, for which we provide groundwork in this paper.
Applications include new inverse theorems for Gowers norms on any finite
abelian group. These theorems are purely group theoretic in that the
correlating harmonics are based on double-coset spaces. This yields progress
towards the Jamneshan-Tao conjecture.
