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:
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