Finite covers and strict boundary slopes of cusped hyperbolic 3-manifolds

TL;DR

This paper shows that two cusped hyperbolic 3-manifolds with isomorphic profinite completions of their fundamental groups have identical A-polynomials and matching boundary slopes, revealing deep algebraic-topological connections.

Contribution

It establishes a link between profinite completions of fundamental groups and geometric invariants like A-polynomials and boundary slopes in hyperbolic 3-manifolds.

Findings

Matching profinite completions imply identical A-polynomials

Boundary slopes are strongly detected and correspond between manifolds

Proves a new algebraic-topological correspondence in hyperbolic geometry

Abstract

We prove that if two cusped hyperbolic $3$-manifolds admit a regular isomorphism between the profinite completions of their fundamental groups, then they share the same $A$-polynomial and their strongly detected boundary slopes match up.