Escape speed of stellar clusters from multiple-generation black-hole mergers in the upper mass gap

Pair instabilities in supernovae might prevent the formation of black holes with masses between $\sim 50 M_\odot$ and $\sim 130 M_\odot$. Multiple generations of black-hole mergers provide a possible way to populate this "mass gap" from below. However this requires an astrophysical environment with a sufficiently large escape speed to retain merger remnants, and prevent them from being ejected by gravitational-wave recoils. We show that, if the mass gap is indeed populated by multiple mergers, the observation of a single black-hole binary component in the mass gap implies that its progenitors grew in an environment with escape speed $v_{\rm esc} \gtrsim 50$ km/s. This is larger than the escape speeds of most globular clusters, requiring denser and heavier environments such as nuclear star clusters or disks-assisted migration in galactic nuclei. A single detection in the upper mass gap would hint at the existence of a much larger population of first-generation events from the same environment, thus providing a tool to disentangle the contribution of different formation channels to the observed merger rate.