A notion of geometric complexity and its application to topological rigidity

2010/08/04 by Erik Guentner, Romain Tessera, Guentner, Erik +3 · 3 citations
Mathematics · #57-xx #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1008.0884

openalex publication_date 2010/08/04 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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 Rⁿ is homeomorphic to N x Rⁿ, 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.

Citations

• Curvature, tangentiality, and controlled topology

Cited by

• The twisted coarse Baum--Connes conjecture and relative hyperbolic groups
• Aspherical groups and manifolds with extreme properties
• Asymptotic Property C for Certain Wreath-like Products of Groups

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