2011/11/03 by Keith Jones, K. L. Jones, Jones, Keith
Computer Science · Mathematics · #20F38 #20F69 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GR #msc:20F38 #msc:20F69
paper · pdf · doi:10.48550/arxiv.1111.0871
13 pages, 1 figure. To be published in Pacific Journal of Mathematics
arxiv created 2011/11/03 · openalex publication_date 2011/11/03 · arxiv updated 2011/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given an action by a finitely generated group G on a locally finite tree T, we view points of the visual boundary \partialT as directions in T and use ρ to lift this sense of direction to G. For each point E ∈ \partialT, this allows us to ask if G is (n - 1)-connected "in the direction of E". The invariant Σn(ρ) ⊆ \partialT then records the set of directions in which G is (n-1)-connected. In this paper, we introduce a family of actions for which Σ1(ρ) can be calculated through analysis of certain quotient maps between trees. We show that for actions of this sort, under reasonable hypotheses, Σ1(ρ) consists of no more than a single point. By strengthening the hypotheses, we are able to characterize precisely when a given end point lies in Σn(ρ) for any n.