zbMATH — the first resource for mathematics

All flat manifolds are cusps of hyperbolic orbifolds. (English) Zbl 0998.57038
The authors prove that for every $$n \geq 2$$, the diffeomorphism class of every flat $$n$$-manifold has a representative $$W$$ which arises as some cusp cross-section of a finite volume cusped hyperbolic $$(n+1)$$-orbifold. This is a partial answer to a question posed by Farrell and Zdravkovska which asked if the diffeomorphism class of every flat $$n$$-manifold has a representative $$W$$ which arises as the cusp cross-section of a finite volume $$1$$-cusped hyperbolic $$n$$-manifold. However, the authors have been unable to pass to manifolds from orbifolds, or to guarantee that their examples are $$1$$-cusped. The main ingredient of their proof is arithmetic, the main technical result being that the fundamental group of a flat $$n$$-manifold embeds as a subgroup of $$O_{0}(q_{n+2};{\mathbb Z})$$, for some quadratic form $$q_{n+2}$$ defined over $${\mathbb Q}$$ of signature $$(n+1,1)$$. This together with some separability arguments allows the authors to prove the main result.

MSC:
 57M50 General geometric structures on low-dimensional manifolds 57R99 Differential topology
Full Text:
References:
 [1] I Agol, D D Long, A W Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. $$(2)$$ 153 (2001) 599 · Zbl 1067.20067 [2] A Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963) 111 · Zbl 0116.38603 [3] A Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. $$(2)$$ 75 (1962) 485 · Zbl 0107.14804 [4] L S Charlap, Bieberbach groups and flat manifolds, Universitext, Springer (1986) · Zbl 0608.53001 [5] F T Farrell, S Zdravkovska, Do almost flat manifolds bound?, Michigan Math. J. 30 (1983) 199 · Zbl 0543.53037 [6] M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231 · Zbl 0432.53020 [7] G C Hamrick, D C Royster, Flat Riemannian manifolds are boundaries, Invent. Math. 66 (1982) 405 · Zbl 0468.57025 [8] T Y Lam, The algebraic theory of quadratic forms, W. A. Benjamin, Reading, MA (1973) · Zbl 0259.10019 [9] D D Long, Immersions and embeddings of totally geodesic surfaces, Bull. London Math. Soc. 19 (1987) 481 · Zbl 0596.57011 [10] D D Long, A W Reid, On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topol. 4 (2000) 171 · Zbl 0961.57011 [11] B E Nimershiem, All flat three-manifolds appear as cusps of hyperbolic four-manifolds, Topology Appl. 90 (1998) 109 · Zbl 0927.57010 [12] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994) · Zbl 0809.51001 [13] J G Ratcliffe, S T Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000) 101 · Zbl 0963.57012 [14] W P Thurston, Three-dimensional geometry and topology Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997) · Zbl 0873.57001 [15] È B Vinberg, The groups of units of certain quadratic forms, Mat. Sb. $$($$N.S.$$)$$ 87(129) (1972) 18 · Zbl 0252.20054 [16] È B Vinberg, O V Shvartsman, Discrete groups of motions of spaces of constant curvature, Encyclopaedia Math. Sci. 29, Springer (1993) 139 · Zbl 0787.22012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.