Geometric finiteness in large families in dimension 3

We prove a diffeomorphism type finiteness theorem in dimension 3 for families C covered by finitely many distance-like functions with bounded twist. C contains families of bounded isoembolic volume and twist. However, the subfamilies of C with fixed volume are not Hausdorff precompact. Our methods do not involve Hausdorff limits and they yield explicit estimates on upper bounds for the number of diffeomorphism types.