Spanning trees in hyperbolic graphs

TL;DR

This paper constructs spanning trees in hyperbolic graphs that effectively represent their boundary structure, with bounds depending on the boundary's Assouad dimension, refining Gromov's earlier results.

Contribution

It introduces a method to build spanning trees in hyperbolic graphs with controlled boundary rays, improving the understanding of hyperbolic boundary representations.

Findings

Bounded number of rays to each boundary point depending on boundary dimension

Construction of a spanning tree with boundary properties within the hyperbolic graph

Sharpened Gromov boundary surjection with finite-to-one property

Abstract

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.