Aspherical groups and manifolds with extreme properties

2011/03/20 by Mark Sapir, Sapir, Mark
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1103.3873

openalex publication_date 2011/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold M such that the fundamental group π₁(M) coarsely contains an expander, and so it has infinite asymptotic dimension, is not coarsely embeddable into a Hilbert space, and does not satisfy the Baum-Connes conjecture with coefficients. Closed aspherical manifolds with any of these properties were previously unknown.

Citations

• Survey on aspherical manifolds
• A notion of geometric complexity and its application to topological rigidity

Related

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• Aspherical 4-manifolds with elementary amenable fundamental group
• Survey on aspherical manifolds
• Fundamental groups of aspherical manifolds that collapse
• Aspherical manifolds with boundary