Minimal complexity cusped hyperbolic 3-manifolds with geodesic boundary

In the early 2000s, Frigerio, Martelli, and Petronio studied $3$-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class $M_{g,k}$ of smallest complexity manifolds having $k$ torus cusps and connected totally geodesic boundary a surface of genus $g$. In this paper, we provide a complete classification of the manifolds in $M_{k,k}$ and $M_{k+1,k}$, which are the cases when the genus $g$ is as small as possible. In addition to classifying manifolds in $M_{k,k}$, $M_{k+1,k}$, we describe their isometry groups as well as a relationship between these two sets via Dehn filling on small slopes. Finally, we give a description of important commensurability invariants of the manifolds in $M_{k,k}$.