Duke Mathematical Journal

Non-LERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

Hongbin Sun

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We will show that for any noncompact arithmetic hyperbolic m-manifold with m>3, and any compact arithmetic hyperbolic m-manifold with m>4 that is not a 7-dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3-manifold groups. We will also show that a compact orientable irreducible 3-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

Article information

Duke Math. J., Volume 168, Number 4 (2019), 655-696.

Received: 2 March 2018
Revised: 27 August 2018
First available in Project Euclid: 4 February 2019

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Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus
Secondary: 57M05: Fundamental group, presentations, free differential calculus 20E26: Residual properties and generalizations; residually finite groups 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]

locally extended residually finite graph of groups hyperbolic 3-manifolds arithmetic hyperbolic manifolds


Sun, Hongbin. Non-LERFness of arithmetic hyperbolic manifold groups and mixed $3$ -manifold groups. Duke Math. J. 168 (2019), no. 4, 655--696. doi:10.1215/00127094-2018-0048. https://projecteuclid.org/euclid.dmj/1549270813

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