The smallest hyperbolic 6-manifolds

Everitt, Brent; Ratcliffe, John; Tschantz, Steven

doi:10.1090/S1079-6762-05-00145-9

The smallest hyperbolic 6-manifolds

Authors: Brent Everitt, John Ratcliffe and Steven Tschantz
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 40-46
MSC (1991): Primary 57M50
DOI: https://doi.org/10.1090/S1079-6762-05-00145-9
Published electronically: May 27, 2005
MathSciNet review: 2150943
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Abstract: By gluing together copies of an all right-angled Coxeter polytope a number of open hyperbolic $6$-manifolds with Euler characteristic $-1$ are constructed. They are the first known examples of hyperbolic $6$-manifolds having the smallest possible volume.

Colin C. Adams, The noncompact hyperbolic $3$-manifold of minimal volume, Proc. Amer. Math. Soc. 100 (1987), no. 4, 601–606. MR 894423, DOI https://doi.org/10.1090/S0002-9939-1987-0894423-8

Belolipetsky02 M. Belolipetsky, On volumes of arithmetic quotients of $SO(1,n)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear).

Richard Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133–153. MR 913200, DOI https://doi.org/10.1016/0021-8693%2887%2990245-6
Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI https://doi.org/10.2307/1970210
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
I. M. Chiswell, The Euler characteristic of graph products and of Coxeter groups, Discrete groups and geometry (Birmingham, 1991) London Math. Soc. Lecture Note Ser., vol. 173, Cambridge Univ. Press, Cambridge, 1992, pp. 36–46. MR 1196914, DOI https://doi.org/10.1017/CBO9780511565793.006
Ian M. Chiswell, Euler characteristics of groups, Math. Z. 147 (1976), no. 1, 1–11. MR 396785, DOI https://doi.org/10.1007/BF01214269

Conder04 M. D. E. Conder and C. Maclachlan, Small volume compact hyperbolic $4$-manifolds, to appear, Proc. Amer. Math. Soc.

J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1993. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 1194619
J. H. Conway and N. J. A. Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 233–258. MR 684311, DOI https://doi.org/10.1098/rspa.1982.0157
H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588–621. MR 1503182, DOI https://doi.org/10.2307/1968753
Michael W. Davis, A hyperbolic $4$-manifold, Proc. Amer. Math. Soc. 93 (1985), no. 2, 325–328. MR 770546, DOI https://doi.org/10.1090/S0002-9939-1985-0770546-7
Brent Everitt, Coxeter groups and hyperbolic manifolds, Math. Ann. 330 (2004), no. 1, 127–150. MR 2091682, DOI https://doi.org/10.1007/s00208-004-0543-0

Everitt02b B. Everitt, $3$-Manifolds from Platonic solids, Top. App. 138 (2004), 253–263. 2035484 (2004m:57031)

Ratcliffe05 B. Everitt, J. Ratcliffe and S. Tschantz, Arithmetic hyperbolic $6$-manifolds of smallest volume, (in preparation).

S. V. Matveev and A. T. Fomenko, Isoenergetic surfaces of Hamiltonian systems, the enumeration of three-dimensional manifolds in order of growth of their complexity, and the calculation of the volumes of closed hyperbolic manifolds, Uspekhi Mat. Nauk 43 (1988), no. 1(259), 5–22, 247 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 1, 3–24. MR 937017, DOI https://doi.org/10.1070/RM1988v043n01ABEH001554
Chun Cao and G. Robert Meyerhoff, The orientable cusped hyperbolic $3$-manifolds of minimum volume, Invent. Math. 146 (2001), no. 3, 451–478. MR 1869847, DOI https://doi.org/10.1007/s002220100167
John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. MR 0506372
A. Neumaier and J. J. Seidel, Discrete hyperbolic geometry, Combinatorica 3 (1983), no. 2, 219–237. MR 726460, DOI https://doi.org/10.1007/BF02579296
István Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (1998), no. 2, 497–515. MR 1642716
John G. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics, vol. 149, Springer-Verlag, New York, 1994. MR 1299730
John G. Ratcliffe and Steven T. Tschantz, Volumes of integral congruence hyperbolic manifolds, J. Reine Angew. Math. 488 (1997), 55–78. MR 1465367
John G. Ratcliffe and Steven T. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000), no. 1, 101–125. MR 1758804
John G. Ratcliffe and Steven T. Tschantz, Integral congruence two hyperbolic 5-manifolds, Geom. Dedicata 107 (2004), 187–209. MR 2110762, DOI https://doi.org/10.1007/s10711-004-8120-y
J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216
Carl Ludwig Siegel, Über die analytische Theorie der quadratischen Formen. II, Ann. of Math. (2) 37 (1936), no. 1, 230–263 (German). MR 1503276, DOI https://doi.org/10.2307/1968694
È. B. Vinberg, Hyperbolic groups of reflections, Uspekhi Mat. Nauk 40 (1985), no. 1(241), 29–66, 255 (Russian). MR 783604
È. B. Vinberg and I. M. Kaplinskaja, The groups $O_{18,1}(Z)$ and $O_{19,1}(Z)$, Dokl. Akad. Nauk SSSR 238 (1978), no. 6, 1273–1275 (Russian). MR 0476640
È. B. Vinberg, The groups of units of certain quadratic forms, Mat. Sb. (N.S.) 87(129) (1972), 18–36 (Russian). MR 0295193

Weeks85 J. Weeks, Hyperbolic structures on $3$-manifolds, Ph.D. thesis, Princeton University, 1985.