Geometrically finite amalgamations of hyperbolic 3‐manifold groups are not LERF

DOI10.1112/PLMS.12182zbMATH Open1454.20062arXiv1705.03498OpenAlexW3126100467MaRDI QIDQ4628460

Hongbin Sun

Publication date: 13 March 2019

Published in: Proceedings of the London Mathematical Society (Search for Journal in Brave)

Abstract: We prove that, for any two finite volume hyperbolic $3$-manifolds, the amalgamation of their fundamental groups along any nontrivial geometrically finite subgroup is not LERF. This generalizes the author's previous work on nonLERFness of amalgamations of hyperbolic $3$-manifold groups along abelian subgroups. A consequence of this result is that closed arithmetic hyperbolic $4$-manifolds have nonLERF fundamental groups. Along with the author's previous work, we get that, for any arithmetic hyperbolic manifold with dimension at least $4$, with possible exceptions in $7$-dimensional manifolds defined by the octonion, its fundamental group is not LERF.

Full work available at URL: https://arxiv.org/abs/1705.03498

zbMATH Keywords

amalgamations of 3-manifold groupsarithmetic 4-manifold groupshyerbolic manifoldsLERF property

Mathematics Subject Classification ID

Geometric group theory (20F65) Discrete subgroups of Lie groups (22E40) General geometric structures on low-dimensional manifolds (57M50) Fundamental group, presentations, free differential calculus (57M05) Residual properties and generalizations; residually finite groups (20E26)

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