2013/12/11 by K. L. Jones, Jones, Keith, Keith Jones
Computer Science · Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR
paper · pdf · doi:10.48550/arxiv.1312.3374
openalex publication_date 2013/12/11 · arxiv created 2013/12/12 · arxiv updated 2013/12/13 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant Σ1 by defining Σ1(ρ), ρ an isometric action by a finitely generated group G on a proper CAT(0) space M. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of Σ1(ρ) when ρ is a cocompact action by G = B \rtimes A, A a finitely generated group, on a locally finite Bass-Serre tree T for A. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endpoint from, Σ1(ρ). When A is a finitely generated free group acting on its Cayley graph, we can restate this theorem from a more algebraic perspective, which leads to some general results on Σ1 for such actions.