Algebraic & Geometric Topology

Nonarithmetic hyperbolic manifolds and trace rings

Olivier Mila

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We give a sufficient condition on the hyperplanes used in the Belolipetsky–Thomson inbreeding construction to obtain nonarithmetic manifolds. We explicitly construct infinitely many examples of such manifolds that are pairwise noncommensurable and estimate their volume.

Article information

Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4359-4373.

Received: 1 June 2018
Revised: 23 July 2018
Accepted: 3 August 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 57M50: Geometric structures on low-dimensional manifolds 51M25: Length, area and volume [See also 26B15] 20G30: Linear algebraic groups over global fields and their integers

hyperbolic manifold nonarithmetic lattice trace rings


Mila, Olivier. Nonarithmetic hyperbolic manifolds and trace rings. Algebr. Geom. Topol. 18 (2018), no. 7, 4359--4373. doi:10.2140/agt.2018.18.4359.

Export citation


  • I Agol, Systoles of hyperbolic $4$–manifolds, preprint (2006)
  • M V Belolipetsky, S A Thomson, Systoles of hyperbolic manifolds, Algebr. Geom. Topol. 11 (2011) 1455–1469
  • N Bergeron, T Gelander, A note on local rigidity, Geom. Dedicata 107 (2004) 111–131
  • A Borel, G Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989) 119–171
  • E D Davis, Overrings of commutative rings, II: Integrally closed overrings, Trans. Amer. Math. Soc. 110 (1964) 196–212
  • T Gelander, A Levit, Counting commensurability classes of hyperbolic manifolds, Geom. Funct. Anal. 24 (2014) 1431–1447
  • M Gromov, I Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. 66 (1988) 93–103
  • C Maclachlan, A W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Mathematics 219, Springer (2003)
  • G A Margulis, E B Vinberg, Some linear groups virtually having a free quotient, J. Lie Theory 10 (2000) 171–180
  • M Newman, Integral matrices, Pure and Applied Mathematics 45, Academic, New York (1972)
  • G Prasad, A S Rapinchuk, Weakly commensurable arithmetic groups and isospectral locally symmetric spaces, Publ. Math. Inst. Hautes Études Sci. 109 (2009) 113–184
  • J Raimbault, A note on maximal lattice growth in ${\rm SO}(1,n)$, Int. Math. Res. Not. 2013 (2013) 3722–3731
  • J G Ratcliffe, Foundations of hyperbolic manifolds, 2nd edition, Graduate Texts in Mathematics 149, Springer (2006)
  • J G Ratcliffe, S T Tschantz, Volumes of integral congruence hyperbolic manifolds, J. Reine Angew. Math. 488 (1997) 55–78
  • S Thomson, Quasi-arithmeticity of lattices in $\mathrm{PO}(n,1)$, Geom. Dedicata 180 (2016) 85–94
  • E B Vinberg, Discrete groups generated by reflections in Lobachevskii spaces, Mat. Sb. 72 (1967) 471–488 Correction in 73 (1967) 303–303 In Russian; translated in Math. USSR-Sb. 1 (1967) 429–444
  • E B Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 45–55 In Russian; translated in Math. USSR-Izv. 5 (1971) 45–55
  • E B Vinberg, V V Gorbatsevich, O V Shvartsman, Discrete subgroups of Lie groups, from “Lie groups and Lie algebras, II” (E B Vinberg, editor), Encyclopaedia Math. Sci. 21, Springer (2000) 1–123, 217–223