Many cusped hyperbolic 3-manifolds do not bound geometrically

In this note we show that there exist cusped hyperbolic 3 3 -manifolds that embed geodesically but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4 4 -manifolds and by Kolpakov, Reid, and Slavich on embedding arithmetic hyperbolic manifolds.