The truncation of the disk of NGC 4565: Detected up to z=4 kpc, with star formation, and affected by the warp

Context: The hierarchical model of galaxy formation suggests that galaxies are continuously growing. However, our position inside the Milky Way prevents us from studying the disk edge. Truncations are low surface brightness features located in the disk outskirts of external galaxies. They indicate where the disk brightness abruptly drops and their location is thought to change dynamically. In previous analyses of Milky Way-like galaxies, truncations were detected up to 3 kpc above the mid-plane but whether they remain present beyond that height remains unclear. Aims: Our goal is to determine whether truncations can be detected above 3 kpc height in the Milky Way-like galaxy NGC 4565, thus establishing the actual disk thickness. We also aim to study how the truncation relates to disk properties such as star formation activity or the warp. Methods: We perform a vertical study of the disk of NGC 4565 edge in unprecedented detail. We explore the truncation radius at different heights above/below the disk mid-plane (0<z<8 kpc) and at different wavelengths. We use new ultra-deep optical data ($\mu_{g,\rm{lim}}=30.5$ mag arcsec$^{-2}$; $3 \sigma$ within $10 \times 10$ arcsec$^{2}$ boxes) in the $g$, $r$ and $i$ broad bands, along with near- and far-ultraviolet, H$\alpha$, and \ion{H}{i} observations. Results: We detect the truncation up to 4 kpc in the $g$, $r$ and $i$ ultra-deep bands which is 1 kpc higher than in any previous study for any galaxy. The radial position of the truncation remains constant up to 3 kpc while higher up it is located at a smaller radius. This result is independent of the wavelength but is affected by the presence of the warp. Conclusions: We propose an inside-out growth scenario for the formation of the disk of NGC 4565. Our results point towards the truncation feature being linked to a star-forming threshold and to the onset of the disk warp.