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

Paul Rapoport

TL;DR

The work addresses the problem of distinguishing 3-manifold groups via profinite data in the Dehn-filling landscape. It combines SL$(2,\mathbb{C})$ representation theory and model-theoretic transfer (via Lefschetz principles) to relate complex character varieties to their finite-field analogues, using the Sky Road to connect representations with topological surfaces. The central result shows that, for a residually finite group $\Gamma$ with finite complex character variety, all but finitely many hyperbolic Dehn fillings $M_{m/n}$ yield fundamental groups whose profinite completions are not isomorphic to $\widehat{\Gamma}$, advancing relative profinite rigidity in 3-manifold groups. The approach provides a broadly applicable framework that links profinite invariants to representation data across characteristics, with potential implications for absolute rigidity and further modeling of 3-manifold group distinctions.

Abstract

We use model theory to study relative profinite rigidity of $3$-manifold groups and show that given any residually finite group $Γ$ with finite character variety and single-cusped finite volume hyperbolic $3$-manifold $M$, cofinitely many Dehn fillings $M_{p/q}$ are profinitely distinguishable from $Γ$.

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

TL;DR

The work addresses the problem of distinguishing 3-manifold groups via profinite data in the Dehn-filling landscape. It combines SL representation theory and model-theoretic transfer (via Lefschetz principles) to relate complex character varieties to their finite-field analogues, using the Sky Road to connect representations with topological surfaces. The central result shows that, for a residually finite group with finite complex character variety, all but finitely many hyperbolic Dehn fillings yield fundamental groups whose profinite completions are not isomorphic to , advancing relative profinite rigidity in 3-manifold groups. The approach provides a broadly applicable framework that links profinite invariants to representation data across characteristics, with potential implications for absolute rigidity and further modeling of 3-manifold group distinctions.

Abstract

We use model theory to study relative profinite rigidity of -manifold groups and show that given any residually finite group with finite character variety and single-cusped finite volume hyperbolic -manifold , cofinitely many Dehn fillings are profinitely distinguishable from .

Paper Structure

This paper contains 13 sections, 36 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic Geometry and Dehn Surgery
  4. Representation Theory
  5. The Sky Road and Contrapositive
  6. Profinite Groups
  7. Motivating Questions
  8. Model Theory And Its Uses
  9. Main Result
  10. More Model Theory
  11. Constraints on Profinite Completions
  12. Main Theorem
  13. Future Work

Key Result

Theorem 1

Let $\Gamma$ be any finitely generated residually finite group with $|\chi^I_\mathbb{C}(\Gamma)| < \infty$, and let $M$ be an oriented finite-volume hyperbolic 3-manifold with a single cusp. Then for hyperbolic Dehn fillings with all but finitely many choices of surgery coefficient $M_{m/n}$ with $\

Theorems & Definitions (84)

  • Theorem 1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Remark
  • ...and 74 more