Discrete groups without finite quotients

Cremaschi, Tommaso; Souto, Juan

doi:10.1016/j.topol.2018.08.013

Cited by 7 publications

(8 citation statements)

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“…Examples include Pride's group, which was already discussed in Example 2.13, as well as certain finitely presented groups such as Higman's group [Hig51] or Burger-Mozes groups [BM97]. Recently, more examples have been found in [CS18] among discrete subgroups of Isom(H 3 ).…”

Section: Homomorphisms Onto Metric Quotientsmentioning

confidence: 99%

Ultrametric analogues of Ulam stability of groups

Fournier-Facio¹

2021

Preprint

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We study stability of metric approximations of countable groups with respect to groups endowed with ultrametrics, the main case study being a p-adic analogue of Ulam stability, where we take GL n (Z p ) as approximating groups instead of U(n). For finitely presented groups, the ultrametric nature implies equivalence of the pointwise and uniform stability problems, and the profinite one implies that the corresponding approximation property is equivalent to residual finiteness. Moreover, a group is uniformly stable if and only if its largest residually finite quotient is. We provide several examples of uniformly stable groups: these include finite groups, virtually free groups, some groups acting on rooted trees, and certain lamplighter and (Generalized) Baumslag-Solitar groups.

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Section: Homomorphisms Onto Metric Quotientsmentioning

confidence: 99%

Ultrametric analogues of Ulam stability of groups

Fournier-Facio¹

2021

Preprint

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“…Proposition 5. 9. Let N be a Seifert 3-manifold whose base orbifold is an orientable surface with positive genus and nontrivial boundary, L < π 1 (N ) be a finitely generated infinitely index subgroup that contains a power of the regular fiber, and K ⊂ N L be any compact subset.…”

Section: Proofmentioning

confidence: 99%

“…At first, it follows from and the geometrization of 3‐manifolds (, see also ) that all finitely generated 3‐manifold groups are residually finite, that is, the trivial subgroup is separable. For a recent work on constructing infinitely generated 3‐manifold groups with nonresidually finite fundamental groups, see .…”

Section: Preliminarymentioning

confidence: 99%

A characterization on separable subgroups of 3‐manifold groups

Sun

2019

Journal of Topology

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In this paper, we give a complete characterization on which finitely generated subgroups of finitely generated 3‐manifold groups are separable. Our characterization generalizes Liu's spirality character on surface subgroups of closed 3‐manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and ∂‐irreducible 3‐manifold M with nontrivial torus decomposition, π1false(Mfalse) is locally extended residually finite if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2.

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“…Then, log log C g,n, 4 arcsinh(1) χ 2 + coth π 12 χ + π sinh 1 2 arcsinh tanh(π/12) κ . We expect these effective bounds might be applicable to the study of the deformation spaces of certain infinite-type hyperbolic 3-manifolds introduced by the first named author, [6,8,9] as explained at the end of the paper.…”

Section: Introductionmentioning

confidence: 99%