Complete cohomogeneity one hypersurfaces of $\mathbb{H}^{n+1}$

We study isometric immersions $f: M^n \rightarrow \mathbb{H}^{n+1}$ into hyperbolic space of dimension $n+1$ of a complete Riemannian manifold of dimension $n$ on which a compact connected group of intrinsic isometries acts with principal orbits of codimension one. We provide a characterization if either $n \geq 3$ and $M^n$ is compact, or $n \geq 5$ and the connected components of the set where the sectional curvature is constant and equal to $-1$ are bounded.