  We study compact three-manifolds with boundary obtained by randomly gluing
together truncated tetrahedra along their faces. We prove that, asymptotically
almost surely as the number of tetrahedra tends to infinity, these manifolds
are connected and have a single boundary component. We prove a law of large
numbers for the genus of this boundary component, we show that the Heegaard
genus of these manifolds is linear in the number of tetrahedra and we bound
their first Betti number.
  We also show that, asymptotically almost surely as the number of tetrahedra
tends to infinity, our manifolds admit a unique hyperbolic metric with totally
geodesic boundary. We prove a law of large numbers for the volume of this
metric, prove that the associated Laplacian has a uniform spectral gap and show
that the diameter of our manifolds is logarithmic as a function of their
volume. Finally, we determine the Benjamini--Schramm limit of our sequence of
random manifolds.
