An inverse theorem for all finite abelian groups via nilmanifolds

We prove a first inverse theorem for Gowers norms on all finite abelian groups that uses only nilmanifolds (rather than possibly more general nilspaces). This makes progress toward confirming the Jamneshan--Tao conjecture. The correlating function in our theorem is a projected nilsequence, obtained as the fiber-wise average of a nilsequence defined on a boundedly-larger abelian group extending the original abelian group. This result is tight in the following sense: we prove also that $k$-step projected nilsequences of bounded complexity are genuine obstructions to having small Gowers $U^{k+1}$-norm. This inverse theorem relies on a new result concerning compact finite-rank (CFR) nilspaces, which is the main contribution in this paper: every $k$-step CFR nilspace is a factor of a $k$-step nilmanifold. This new connection between the classical theory of nilmanifolds and the more recent theory of nilspaces has applications beyond arithmetic combinatorics. We illustrate this with an application in topological dynamics, by proving the following result making progress on a question of Jamneshan, Shalom and Tao: every minimal $\mathbb{Z}^\omega$-system of order $k$ is a factor of an inverse limit of $\mathbb{Z}^\omega$-polynomial orbit systems of order $k$, these being natural generalizations of nilsystems alternative to translational systems.