Spanning trees in hyperbolic graphs

Details Spanning trees in hyperbolic graphs Spanning trees in hyperbolic graphs

2009-10-29

arxiv.org/abs/0910.5605v1

Mathematics

Combinatorics

16 pages

Scientific paper

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.

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