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Geometrically bounding $3$–manifolds, volume and Betti numbers

Jiming Ma and Fangting Zheng

Algebraic & Geometric Topology 23 (2023) 1055–1096

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 3. 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).

hyperbolic 3-manifolds, geometrically bounding, hyperbolic 4-manifolds, small cover
Mathematical Subject Classification
Primary: 57R90, 57M50, 57S25
Supplementary material

Figures with supplemental calculations

Received: 9 March 2020
Revised: 16 November 2020
Accepted: 30 December 2020
Published: 6 June 2023
Jiming Ma
School of Mathematical Sciences
Fudan University
Fangting Zheng
Department of Pure Mathematics
Xi’an Jiaotong-Liverpool University

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