Geometry & Topology

Noncoherence of arithmetic hyperbolic lattices

Michael Kapovich

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We prove that all arithmetic lattices in O(n,1), n4, n7, are noncoherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n2, and of uniform lattices in SU(2,1) which have infinite abelianization.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 39-71.

Received: 7 September 2011
Revised: 2 September 2012
Accepted: 12 July 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F06: Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups

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Kapovich, Michael. Noncoherence of arithmetic hyperbolic lattices. Geom. Topol. 17 (2013), no. 1, 39--71. doi:10.2140/gt.2013.17.39.

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  • I Agol, Systoles of hyperbolic $4$–manifolds
  • I Agol, The virtual Haken conjecture
  • I Agol, Criteria for virtual fibering, J. Topol. 1 (2008) 269–284
  • I Agol, D D Long, A W Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. 153 (2001) 599–621
  • G Baumslag, J E Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. 30 (1984) 44–52
  • M Bestvina, N Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997) 445–470
  • R Bieri, Homological dimension of discrete groups, 2nd edition, Queen Mary College Mathematical Notes, Queen Mary College Department of Pure Mathematics, London (1981)
  • J S Birman, A Lubotzky, J McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983) 1107–1120
  • B H Bowditch, G Mess, A $4$–dimensional Kleinian group, Trans. Amer. Math. Soc. 344 (1994) 391–405
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer, Berlin (1999)
  • K Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. 135 (1992) 165–182
  • P Deligne, G D Mostow, Commensurabilities among lattices in ${\mathrm PU}(1,n)$, Annals of Mathematics Studies 132, Princeton Univ. Press (1993)
  • T Delzant, L'invariant de Bieri–Neumann–Strebel des groupes fondamentaux des variétés kählériennes, Math. Ann. 348 (2010) 119–125
  • R Gitik, Doubles of groups and hyperbolic LERF $3$–manifolds, Ann. of Math. 150 (1999) 775–806
  • R Gitik, Ping-pong on negatively curved groups, J. Algebra 217 (1999) 65–72
  • M Gromov, R Schoen, Harmonic maps into singular spaces and $p$–adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. (1992) 165–246
  • F J Grunewald, On some groups which cannot be finitely presented, J. London Math. Soc. 17 (1978) 427–436
  • F Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008) 167–209
  • F Haglund, J Świątkowski, Separating quasi-convex subgroups in $7$–systolic groups, Groups Geom. Dyn. 2 (2008) 223–244
  • M Hall, Jr, Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949) 421–432
  • J A Hillman, Private communication (1998)
  • J A Hillman, Complex surfaces which are fibre bundles, Topology Appl. 100 (2000) 187–191
  • J A Hillman, Four-manifolds, geometries and knots, volume 5 of Geometry & Topology Monographs (2002)
  • J A Hillman, D H Kochloukova, Finiteness conditions and ${\rm PD}\sb r$–group covers of ${\rm PD}\sb n$–complexes, Math. Z. 256 (2007) 45–56
  • M Kapovich, On normal subgroups in the fundamental groups of complex surfaces (1998)
  • M Kapovich, Representations of polygons of finite groups, Geom. Topol. 9 (2005) 1915–1951 (electronic)
  • M Kapovich, L Potyagailo, On the absence of Ahlfors' finiteness theorem for Kleinian groups in dimension three, Topology Appl. 40 (1991) 83–91
  • M Kapovich, L Potyagailo, E Vinberg, Noncoherence of some lattices in ${\rm Isom}(\mathbb H^n)$, from: “The Zieschang Gedenkschrift”, (M Boileau, M Scharlemann, R Weidmann, editors), Geom. Topol. Monogr. 14, Geom. Topol. Publ., Coventry (2008) 335–351
  • D Kazhdan, Some applications of the Weil representation, J. Analyse Mat. 32 (1977) 235–248
  • J-S Li, J J Millson, On the first Betti number of a hyperbolic manifold with an arithmetic fundamental group, Duke Math. J. 71 (1993) 365–401
  • K Liu, Geometric height inequalities, Math. Res. Lett. 3 (1996) 693–702
  • D D Long, G A Niblo, Subgroup separability and $3$–manifold groups, Math. Z. 207 (1991) 209–215
  • C Maclachlan, A W Reid, The arithmetic of hyperbolic $3$–manifolds, Graduate Texts in Mathematics 219, Springer, New York (2003)
  • D B McReynolds, Arithmetic lattices in $\mathrm{SU}(n, 1)$ (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • V Platonov, A Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Boston, MA (1994)
  • L Potyagaĭlo, The problem of finiteness for Kleinian groups in $3$–space, from: “Knots 90”, (A Kawauchi, editor), de Gruyter, Berlin (1992) 619–623
  • L Potyagaĭlo, Finitely generated Kleinian groups in $3$–space and $3$–manifolds of infinite homotopy type, Trans. Amer. Math. Soc. 344 (1994) 57–77
  • J D Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton Univ. Press (1990)
  • G P Scott, Compact submanifolds of $3$–manifolds, J. London Math. Soc. (2) 7 (1973) 246–250
  • G P Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978) 555–565
  • M Stover, Property (FA) and lattices in ${\mathrm SU}(2,1)$, Internat. J. Algebra Comput. 17 (2007) 1335–1347
  • È B Vinberg, Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sb. 72 (114) (1967) 471–488; correction, ibid. 73 (115), 303
  • È B Vinberg, O V Shvartsman, Discrete groups of motions of spaces of constant curvature, from: “Geometry, II”, Encyclopaedia Math. Sci. 29, Springer, Berlin (1993) 139–248
  • C Wall, List of problems, from: “Homological group theory”, (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 369–394
  • N R Wallach, Square integrable automorphic forms and cohomology of arithmetic quotients of $\mathrm{SU}(p,\,q)$, Math. Ann. 266 (1984) 261–278
  • D T Wise, Subgroup separability of graphs of free groups with cyclic edge groups, Q. J. Math. 51 (2000) 107–129
  • D T Wise, Subgroup separability of the figure $8$ knot group, Topology 45 (2006) 421–463
  • D T Wise, Morse theory, random subgraphs, and incoherent groups, Bull. Lond. Math. Soc. 43 (2011) 840–848
  • D T Wise, The structure of groups with a quasiconvex hierarchy (2012)
  • S-K Yeung, Virtual first Betti number and integrality of compact complex two-ball quotients, Int. Math. Res. Not. 2004 (2004) 1967–1988