Characterizing candidates for Cannon's Conjecture from Geometric Measure Theory

TL;DR

This paper links hyperbolic groups with boundary sphere to 3-manifolds of constant negative curvature via a spherical Plateau problem, providing a new characterization related to Cannon's Conjecture.

Contribution

It establishes a criterion connecting group boundaries, geometric measure theory, and 3-manifold realizations, advancing understanding of Cannon's Conjecture.

Findings

Hyperbolic groups with boundary S^2 relate to 3-manifolds of negative curvature.

Solution to a spherical Plateau problem characterizes such groups as 3-manifold fundamental groups.

Provides a new approach to Cannon's Conjecture through geometric measure theory.

Abstract

We show that recent work of Song implies that torsion-free hyperbolic groups with Gromov boundary $S^2$ are realized as fundamental groups of closed 3-manifolds of constant negative curvature if and only if the solution to an associated spherical Plateau problem for group homology is isometric to such a 3-manifold, and suggest some related questions.