Connectivity properties of group actions on non-positively curved spaces I: Controlled connectivity and openness results

1998/11/03 by Robert Bieri, Bieri, Robert, Ross Geoghegan +1 · 1 citation
Computer Science · Mathematics · #20F32 #57N99 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR #math.GT #msc:20F32 #msc:57N99

paper · pdf · doi:10.48550/arxiv.math/9811007

43 pages

arxiv created 1998/11/03 · openalex publication_date 1998/11/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Isometric actions of G on M are (by definition) points in the space R := Hom(G, Isom(M)) with the compact open topology. Sample theorems: 1. The cocompact actions form an open subset of R. 2. The cocompact actions with discrete orbits whose point-stabilizers have type F_n form an open subset of the subspace of R consisting of all actions with discrete orbits. (F₁ means finitely generated, F₂ means finitely presented etc.) The key idea is to introduce a new "controlled topology" invariant of such actions - dependent on n - which is unfamiliar when the orbits are not discrete but which becomes familiar (cf 2.) when the orbits are discrete. (This is the first of two papers.)

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