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Some hyperbolic three-manifolds that bound geometrically


Authors: Alexander Kolpakov, Bruno Martelli and Steven Tschantz
Journal: Proc. Amer. Math. Soc. 143 (2015), 4103-4111
MSC (2010): Primary 57N16; Secondary 52B11, 52C45
DOI: https://doi.org/10.1090/proc/12520
Published electronically: April 6, 2015
Erratum: Proc. Amer. Math. Soc. 144 (2016) 3647-3648.
MathSciNet review: 3359598
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Abstract: A closed connected hyperbolic $ n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $ (n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension.

We construct here infinitely many explicit examples in dimension $ n=3$ using right-angled dodecahedra and 120-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $ k\geqslant 1$, we build an orientable compact closed 3-manifold tessellated by $ 16k$ right-angled dodecahedra that bounds a 4-manifold tessellated by $ 32k$ right-angled 120-cells.

A notable feature of this family is that the ratio between the volumes of the 4-manifolds and their boundary components is constant and, in particular, bounded.


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Additional Information

Alexander Kolpakov
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto Ontario, M5S 2E4, Canada
Email: kolpakov.alexander@gmail.com

Bruno Martelli
Affiliation: Dipartimento di Matematica “Tonelli”, Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, Italy
Email: martelli@dm.unipi.it

Steven Tschantz
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240
Email: steven.tschantz@vanderbilt.edu

DOI: https://doi.org/10.1090/proc/12520
Keywords: 4-manifold, gravitational instanton, Coxeter group
Received by editor(s): November 12, 2013
Received by editor(s) in revised form: April 3, 2014, and April 4, 2014
Published electronically: April 6, 2015
Additional Notes: The first author was supported by the SNSF researcher scholarship P300P2-151316.
The second author was supported by the Italian FIRB project “Geometry and topology of low-dimensional manifolds”, RBFR10GHHH
Communicated by: Kevin Whyte
Article copyright: © Copyright 2015 American Mathematical Society