2014/03/30 by Brendon LaBuz, LaBuz, Brendon
Computer Science · Mathematics · #20E08 #20F67 #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1403.7805
openalex publication_date 2014/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The set of homotopy classes of based paths in the Hawaiian earring has a natural \mathbb R-tree structure, but under that metric the action by the fundamental group is not by isometries. Following a suggestion by Cannon and Conner, this paper defines an \mathbb Rω-metric that does admit for an isometric action by the fundamental group. The space does not become an \mathbb Rω-tree but is 0-hyperbolic and embeds in an \mathbb Rω-tree. Cannon and Conner define big free groups BF(c) for cardinal number c which are a generalization of the fundamental group of the Hawaiian earring. They define a big Cayley graph which coincides with the set of homotopy classes of paths in the case of the Hawaiian earring. Instead of inserting real intervals to obtain the Cayley graph, we can insert \mathbb Rc-intervals and obtain a new \mathbb Rc-tree which admits an isometric action. In fact we do not need all of \mathbb Rc; we can insert \mathbb Zc-intervals and obtain a \mathbb Zc-tree. In the case of the Hawaiian earring we give a combinatorial description of the \mathbb Zω-tree and the corresponding action.