On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume 3–manifolds

Rapoport, Paul

doi:10.2140/agt.2024.24.4779

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On the profinite distinguishability of hyperbolic Dehn fillings of finite-volume $3$–manifolds

Paul Rapoport

Algebraic & Geometric Topology 24 (2024) 4779–4797

DOI: 10.2140/agt.2024.24.4779

Abstract

We use the Culler–Shalen machine and tools from model theory to study the profinite rigidity of residually finite groups, especially $3$ –manifold groups. We borrow a transfer principle from model theory to apply to $ℂ$ –character varieties in order to study cofinite collections of $𝔽_{p}$ –character varieties and prove that under certain finiteness conditions weaker than non-Hakenness, they all have the same (finite) cardinality. We prove that residually finite groups satisfying a niceness property are almost relatively profinitely distinguishable within a geometrically relevant class, and we finish up by applying that result to knot complements in $S^{3}$ in particular.

Keywords

geometric group theory, profinite, non-Haken, Dehn, filling, rigidity

Mathematical Subject Classification

Primary: 20F65, 57M07

Secondary: 03C07, 03C52

References

Publication

Received: 26 August 2021

Revised: 14 September 2023

Accepted: 1 October 2023

Published: 27 December 2024

Authors

Paul Rapoport
Department of Mathematics Temple University Philadelphia, PA United States

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Volume 24, issue 9 (2024)