Geometrically bounding 3–manifolds, volume and Betti numbers

A hyperbolic 3–manifold is geometrically bounding if it is the only boundary of a totally geodesic hyperbolic 4–manifold. According to previous results of Long and Reid (2000) and Meyerhoff and Neumann (1992), geometrically bounding closed hyperbolic 3–manifolds are very rare. Assume the value v ≈ 4:3062::: for the volume of the regular right-angled hyperbolic dodecahedron P in H³. For each positive integer n and each odd integer k in [1, 5n + 3], we construct a closed hyperbolic 3–manifold M with β¹(_M) = k and vol(M) = 16nv which bounds a totally geodesic hyperbolic 4–manifold. In particular, for every positive odd integer k, there are infinitely many geometrically bounding 3–manifolds whose first Betti numbers are k. The proof exploits the real toric manifold theory over a sequence of stacking dodecahedra, together with some results obtained by Kolpakov, Martelli and Tschantz (2015).