2009/10/29 by Matthias Hamann, Hamann, Matthias
Mathematics · #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.CO
paper · pdf · doi:10.48550/arxiv.0910.5605
18 pages, 1 figure
openalex publication_date 2009/10/29 · arxiv created 2013/01/30 · arxiv updated 2013/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we construct spanning trees in hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has a bounded number of distinct rays to each boundary point. The bound depends only on the (Assouad) dimension of the boundary. As a corollary we sharpen a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree outside the graph with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We will construct a tree with these properties inside the hyperbolic graph, which in addition is also a spanning tree of that graph.