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A notion of geometric complexity and its application to topological rigidity

Authors
  • Guentner, Erik
  • Tessera, Romain
  • Yu, Guoliang
Type
Preprint
Publication Date
Aug 04, 2010
Submission Date
Aug 04, 2010
Source
arXiv
License
Yellow
External links

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.

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