Infinitely many arithmetic hyperbolic rational homology $3$–spheres that bound geometrically
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- by L. Ferrari, A. Kolpakov and A. W. Reid;
- Trans. Amer. Math. Soc. 376 (2023), 1979-1997
- DOI: https://doi.org/10.1090/tran/8816
- Published electronically: November 9, 2022
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Abstract:
In this paper we provide the first examples of arithmetic hyperbolic $3$–manifolds that are rational homology spheres and bound geometrically either compact or cusped hyperbolic $4$–manifolds.References
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Nigel Boston and Jordan S. Ellenberg, Pro-$p$ groups and towers of rational homology spheres, Geom. Topol. 10 (2006), 331–334. MR 2224459, DOI 10.2140/gt.2006.10.331
- V. O. Bugaenko, Groups of automorphisms of unimodular hyperbolic quadratic forms over the ring $\textbf {Z}[(\sqrt {5}+1)/2]$, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (1984), 6–12 (Russian). MR 764026
- Frank Calegari and Nathan M. Dunfield, Automorphic forms and rational homology 3-spheres, Geom. Topol. 10 (2006), 295–329. MR 2224458, DOI 10.2140/gt.2006.10.295
- Ted Chinburg, Eduardo Friedman, Kerry N. Jones, and Alan W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), no. 1, 1–40. MR 1882023
- Suyoung Choi and Hanchul Park, Multiplicative structure of the cohomology ring of real toric spaces, Homology Homotopy Appl. 22 (2020), no. 1, 97–115. MR 4027292, DOI 10.4310/HHA.2020.v22.n1.a7
- Michelle Chu and Alexander Kolpakov, A hyperbolic counterpart to Rokhlin’s cobordism theorem, Int. Math. Res. Not. IMRN 4 (2022), 2460–2483. MR 4381923, DOI 10.1093/imrn/rnaa158
- M. Culler, N. M. Dunfield, M. Goerner, J. R. Weeks, et al., SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://snappy.computop.org.
- Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. MR 1104531, DOI 10.1215/S0012-7094-91-06217-4
- Guillaume Dufour, Notes on right-angled Coxeter polyhedra in hyperbolic spaces, Geom. Dedicata 147 (2010), 277–282. MR 2660580, DOI 10.1007/s10711-009-9454-2
- Vincent Emery, On compact hyperbolic manifolds of Euler characteristic two, Algebr. Geom. Topol. 14 (2014), no. 2, 853–861. MR 3160605, DOI 10.2140/agt.2014.14.853
- Leonardo Ferrari, Alexander Kolpakov, and Leone Slavich, Cusps of hyperbolic 4-manifolds and rational homology spheres, Proc. Lond. Math. Soc. (3) 123 (2021), no. 6, 636–648. MR 4368684, DOI 10.1112/plms.12421
- L. Ferrari and A. Kolpakov, A GitHub repository with ancillary SageMath code, https://github.com/sashakolpakov/dodecahedron-colouring.
- David Gabai, Robert Meyerhoff, and Peter Milley, Minimum volume cusped hyperbolic three-manifolds, J. Amer. Math. Soc. 22 (2009), no. 4, 1157–1215. MR 2525782, DOI 10.1090/S0894-0347-09-00639-0
- Anne Garrison and Richard Scott, Small covers of the dodecahedron and the 120-cell, Proc. Amer. Math. Soc. 131 (2003), no. 3, 963–971. MR 1937435, DOI 10.1090/S0002-9939-02-06577-2
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- W. Hantzsche and H. Wendt, Dreidimensionale euklidische Raumformen, Math. Ann. 110 (1935), no. 1, 593–611 (German). MR 1512956, DOI 10.1007/BF01448045
- Alexander Kolpakov, On the optimality of the ideal right-angled 24-cell, Algebr. Geom. Topol. 12 (2012), no. 4, 1941–1960. MR 2994826, DOI 10.2140/agt.2012.12.1941
- Alexander Kolpakov and Bruno Martelli, Hyperbolic four-manifolds with one cusp, Geom. Funct. Anal. 23 (2013), no. 6, 1903–1933. MR 3132905, DOI 10.1007/s00039-013-0247-2
- Alexander Kolpakov, Bruno Martelli, and Steven Tschantz, Some hyperbolic three-manifolds that bound geometrically, Proc. Amer. Math. Soc. 143 (2015), no. 9, 4103–4111. MR 3359598, DOI 10.1090/proc/12520
- Alexander Kolpakov, Alan W. Reid, and Stefano Riolo, Many cusped hyperbolic 3-manifolds do not bound geometrically, Proc. Amer. Math. Soc. 148 (2020), no. 5, 2233–2243. MR 4078106, DOI 10.1090/proc/14573
- Alexander Kolpakov, Alan W. Reid, and Leone Slavich, Embedding arithmetic hyperbolic manifolds, Math. Res. Lett. 25 (2018), no. 4, 1305–1328. MR 3882165, DOI 10.4310/MRL.2018.v25.n4.a12
- Alexander Kolpakov and Leone Slavich, Hyperbolic 4-manifolds, colourings and mutations, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 163–184. MR 3534970, DOI 10.1112/plms/pdw025
- D. D. Long and A. W. Reid, On the geometric boundaries of hyperbolic $4$-manifolds, Geom. Topol. 4 (2000), 171–178. MR 1769269, DOI 10.2140/gt.2000.4.171
- D. D. Long and A. W. Reid, Constructing hyperbolic manifolds which bound geometrically, Math. Res. Lett. 8 (2001), no. 4, 443–455. MR 1849261, DOI 10.4310/MRL.2001.v8.n4.a5
- J. Ma and F. Zheng, Geometrically bounding $3$–manifolds, volume and Betti number, arXiv:1704.02889.
- C. Maclachlan and A. W. Reid, The arithmetic structure of tetrahedral groups of hyperbolic isometries, Mathematika 36 (1989), no. 2, 221–240 (1990). MR 1045784, DOI 10.1112/S0025579300013097
- Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR 1937957, DOI 10.1007/978-1-4757-6720-9
- Robert Meyerhoff and Daniel Ruberman, Mutation and the $\eta$-invariant, J. Differential Geom. 31 (1990), no. 1, 101–130. MR 1030667
- Leonid Potyagailo and Ernest Vinberg, On right-angled reflection groups in hyperbolic spaces, Comment. Math. Helv. 80 (2005), no. 1, 63–73. MR 2130566, DOI 10.4171/CMH/4
- John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer, Cham, [2019] ©2019. Third edition [of 1299730]. MR 4221225, DOI 10.1007/978-3-030-31597-9
- A. W. Reid, The geometry and topology of arithmetic hyperbolic manifolds of simplest type, The Vinberg Lecture Series, 2021, https://vinberg.combgeo.org/alan-reid/.
- J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, RI, 1966, pp. 33–62. MR 224710
- A. Yu. Vesnin, Right-angled polytopes and three-dimensional hyperbolic manifolds, Uspekhi Mat. Nauk 72 (2017), no. 2(434), 147–190 (Russian, with Russian summary); English transl., Russian Math. Surveys 72 (2017), no. 2, 335–374. MR 3635439, DOI 10.4213/rm9762
- È. B. Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 45–55 (Russian). MR 279206
- È. B. Vinberg and O. V. Shvartsman, Discrete groups of motions of spaces of constant curvature, Geometry, II, Encyclopaedia Math. Sci., vol. 29, Springer, Berlin, 1993, pp. 139–248. MR 1254933, DOI 10.1007/978-3-662-02901-5_{2}
Bibliographic Information
- L. Ferrari
- Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile–Argand 11, 2000 Neuchâtel, Switzerland
- MR Author ID: 1484679
- ORCID: 0000-0003-2329-5975
- Email: leonardocpferrari@gmail.com
- A. Kolpakov
- Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile–Argand 11, 2000 Neuchâtel, Switzerland
- MR Author ID: 774696
- ORCID: 0000-0002-6764-8894
- Email: kolpakov.alexander@gmail.com
- A. W. Reid
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005; and Max-Planck-Insititut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
- MR Author ID: 146355
- ORCID: 0000-0003-0367-9453
- Email: alan.reid@rice.edu, areid@mpim-bonn.mpg.de
- Received by editor(s): May 10, 2022
- Received by editor(s) in revised form: August 24, 2022
- Published electronically: November 9, 2022
- Additional Notes: The first and third authors were supported by the Swiss National Science Foundation, project no. PP00P2–202667. The second author was supported by the National Science Foundation and the Max–Planck–Institut für Mathematik
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 1979-1997
- MSC (2020): Primary 57K32; Secondary 57R40, 57R42
- DOI: https://doi.org/10.1090/tran/8816
- MathSciNet review: 4549697