  It is well known that an arbitrary closed orientable $3$-manifold can be
realized as the unique boundary of a compact orientable $4$-manifold, that is,
any closed orientable $3$-manifold is cobordant to zero. In this paper, we
consider the geometric cobordism problem: a hyperbolic $3$-manifold is
geometrically bounding if it is the only boundary of a totally geodesic
hyperbolic 4-manifold. However, there are very rare geometrically bounding
closed hyperbolic 3-manifolds according to the previous research [11,13]. Let
$v \approx 4.3062\ldots$ be the volume of the regular right-angled hyperbolic
dodecahedron in $\mathbb{H}^{3}$, for each $n \in \mathbb{Z}_{+}$ and each odd
integer $k$ in $[1,5n+3]$, we construct a closed hyperbolic 3-manifold $M$ with
$\beta^1(M)=k$ and $vol(M)=16nv$ that bounds a totally geodesic hyperbolic
4-manifold. The proof uses small cover theory over a sequence of linearly-glued
dodecahedra and some results of Kolpakov-Martelli-Tschantz [9].
