A notion of geometric complexity and its application to topological rigidity

TL;DR

This paper introduces finite decomposition complexity (FDC), a geometric invariant, to analyze topological rigidity of manifolds, proving the stable Borel conjecture for FDC groups and exploring their algebraic properties.

Contribution

It defines FDC as a new geometric invariant and demonstrates its applicability to topological rigidity and group classification, extending the class of groups known to satisfy the stable Borel conjecture.

Findings

FDC groups include all countable subgroups of GL(n,K) and elementary amenable groups.

FDC is closed under subgroups, extensions, and various group constructions.

Manifolds with FDC fundamental groups satisfy the stable Borel conjecture.

Abstract

We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M x R^n is homeomorphic to N x R^n, for n large enough. This statement is known as the stable Borel conjecture. On the other hand, we show that the class of FDC groups includes all countable subgroups of GL(n,K), for any field K, all elementary amenable groups, and is closed under taking subgroups, extensions, free amalgamated products, HNN extensions, and direct unions.