Geometric cheap rebuilding property for residually finite amenable groups of type F∞

Determine whether every residually finite infinite amenable group of type F∞ satisfies the geometric cheap rebuilding property of Abért–Bergeron–Frączyk–Gekhtman (2021).

Background

The authors’ algebraic cheap rebuilding properties are inspired by the geometric cheap rebuilding property, which has strong consequences for homology growth via combination theorems. They prove an algebraic weak version (CWR∞) for amenable groups of type FP∞ and deduce vanishing torsion growth, while the geometric version is known for certain classes (e.g., infinite elementary amenable groups of type FP∞).

Whether the full geometric property holds for general residually finite infinite amenable groups of type F∞ is explicitly stated as unknown.