Characterizing hyperbolic spaces and real trees
By Roberto Frigerio, Alessandro Sisto. 2008
55 citations recorded by the source corpus.
Let X be a geodesic metric space. Gromov proved that there exists ε0 > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε0 · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε0. We also show that if all the triangles $${\Delta \subseteq X}$$ satisfy the Rips condition with constant ε0 · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/fc09217bc595ba2286d2cf8f4be57c151f26dd3e
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Gromov hyperbolicity in Cartesian product graphs (2010, background): “The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [2, 3, 4, 7, 8, 9, 11, 12, 18, 19, 20, 21, 22, 24, 26, 27, 29, 30].”
- Gromov hyperbolic cubic graphs (2012, background): “The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [4, 5, 6, 7, 10, 11, 14, 24, 25, 26, 28, 30, 31, 34, 35, 37, 38, 39, 40, 42].”