On the stable Cannon Conjecture

TL;DR

This paper proves a stable version of the Cannon Conjecture, showing that for certain hyperbolic groups, there exists a related manifold with a simple homotopy equivalence to a product involving the group’s classifying space.

Contribution

It introduces a stable version of the Cannon Conjecture, establishing existence and uniqueness results for manifolds related to hyperbolic groups and their boundaries.

Findings

Existence of a closed manifold homotopy equivalent to N × BG.

Uniqueness of the manifold M when N is aspherical and satisfies Farrell-Jones.

Extension of the Cannon Conjecture to a stable setting.

Abstract

The Cannon Conjecture for a torsionfree hyperbolic group G with boundary homeomorphic to S^2 says that G is the fundamental group of an aspherical closed 3-manifold M. It is known that then M is a hyperbolic 3-manifold. We prove the stable version that for any closed manifold N of dimension greater or equal to 2 there exists a closed manifold M together with a simple homotopy equivalence from M to the cartesian product of N and BG. If N is aspherical and pi_1(N) satisfies the Farrell-Jones Conjecture, then M is unique up to homeomorphism.