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

TL;DR

This paper uses model theory to demonstrate that most Dehn fillings of a finite-volume hyperbolic 3-manifold can be distinguished from a broad class of residually finite groups by their profinite completions.

Contribution

It introduces a novel application of model theory to study the profinite rigidity of 3-manifold groups and shows that many Dehn fillings are uniquely identifiable via their profinite properties.

Findings

Cofinitely many Dehn fillings are profinitely distinguishable from certain residually finite groups.

The study connects model theory with geometric topology and group theory.

Provides new insights into the structure of 3-manifold groups through profinite invariants.

Abstract

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