Hamiltonian group actions in cosymplectic geometry

We develop a theory of Hamiltonian group actions on cosymplectic manifolds. These odd-dimensional manifolds combine a codimension-one symplectic foliation with a distinguished Reeb direction, and arise naturally both in stable Hamiltonian geometry and as critical hypersurfaces of $b$-symplectic manifolds.
Our approach is based on a compact symplectic thickening process: every cosymplectic manifold $(M,\alpha,\beta)$ gives rise to a symplectic manifold $(M\times \mathbb{S}^1,\ \beta + \mathrm{d}\theta\wedge\alpha)$. We prove that Hamiltonian cosymplectic actions lift canonically to Hamiltonian symplectic actions on this symplectic manifold. This provides a systematic bridge between equivariant symplectic geometry and the cosymplectic setting.
Using this bridge, we establish cosymplectic analogues of convexity theorems for torus actions, Delzant theorem for toric actions, ABBV localization, Duistermaat-Heckman formulas, and Kirwan surjectivity. The resulting formulas are not merely formal pullbacks from the symplectic case: the Reeb direction appears explicitly through the factor $\alpha$, and, in the mapping-torus case, the localization and volume formulas are governed by the modular period together with the equivariant geometry of the symplectic fiber. We also explain how these results apply to Hamiltonian geometry on the critical hypersurfaces of $b$-symplectic manifolds.