Finitely generated subgroups of lattices in \(\text{PSL}_2\mathbb C\). (English) Zbl 1220.20018
Summary: Let \(\Gamma\) be a lattice in \(\text{PSL}_2\mathbb C\). The pro-normal topology on \(\Gamma\) is defined by taking all cosets of nontrivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup \(\Delta<\Gamma\) is closed in the pro-normal topology. As a corollary we deduce that if \(H\) is a maximal subgroup of a lattice in \(\text{PSL}_2\mathbb C\), then either \(H\) is of finite index or \(H\) is not finitely generated.
MSC:
| 20E15 | Chains and lattices of subgroups, subnormal subgroups |
| 20E28 | Maximal subgroups |
| 20E07 | Subgroup theorems; subgroup growth |
| 57M07 | Topological methods in group theory |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
| 22E40 | Discrete subgroups of Lie groups |
| 20E18 | Limits, profinite groups |
Keywords:
finitely generated subgroups; pro-normal topology; cosets of normal subgroups; pro-finite topology; closed subgroups; maximal subgroupsReferences:
| [1] | I. Agol. Tameness of hyperbolic 3-manifolds. Preprint available at http://front. math.ucdavis.edu/math.GT/0405568. |
| [2] | I. Agol, D. D. Long, and A. W. Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. (2) 153 (2001), no. 3, 599 – 621. · Zbl 1067.20067 · doi:10.2307/2661363 |
| [3] | Danny Calegari and David Gabai, Shrinkwrapping and the taming of hyperbolic 3-manifolds, J. Amer. Math. Soc. 19 (2006), no. 2, 385 – 446. · Zbl 1090.57010 |
| [4] | Richard D. Canary, A covering theorem for hyperbolic 3-manifolds and its applications, Topology 35 (1996), no. 3, 751 – 778. · Zbl 0863.57010 · doi:10.1016/0040-9383(94)00055-7 |
| [5] | Tsachik Gelander and Yair Glasner, Countable primitive groups, Geom. Funct. Anal. 17 (2008), no. 5, 1479 – 1523. · Zbl 1138.20005 · doi:10.1007/s00039-007-0630-y |
| [6] | Rita Gitik, Doubles of groups and hyperbolic LERF 3-manifolds, Ann. of Math. (2) 150 (1999), no. 3, 775 – 806. · Zbl 0944.20023 · doi:10.2307/121056 |
| [7] | R. I. Grigorchuk and J. S. Wilson, A structural property concerning abstract commensurability of subgroups, J. London Math. Soc. (2) 68 (2003), no. 3, 671 – 682. · Zbl 1063.20033 · doi:10.1112/S0024610703004745 |
| [8] | M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1 – 295. · Zbl 0841.20039 |
| [9] | G. A. Margulis and G. A. Soĭfer, Maximal subgroups of infinite index in finitely generated linear groups, J. Algebra 69 (1981), no. 1, 1 – 23. · Zbl 0457.20046 · doi:10.1016/0021-8693(81)90123-X |
| [10] | Ashot Minasyan, On residual properties of word hyperbolic groups, J. Group Theory 9 (2006), no. 5, 695 – 714. · Zbl 1116.20022 · doi:10.1515/JGT.2006.045 |
| [11] | Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555 – 565. · Zbl 0412.57006 · doi:10.1112/jlms/s2-17.3.555 |
| [12] | W. Thurston. The topology and geometry of 3-manifolds. Princeton Univ. Lecture Notes, 1976-1979. Available from the MSRI website www.msri.org. |
| [13] | William P. Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203 – 246. · Zbl 0668.57015 · doi:10.2307/1971277 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
