A characterization of Gromov hyperbolicity of surfaces with variable negative curvature

Abstract

In this paper we show that, in order to check Gromov hyperbolicity of any surface with curvature K ≤ −k 2 < 0, we just need to verify the Rips condition on a very small class of triangles, namely, those contained in simple closed geodesics. This result is, in fact, a new characterization of Gromov hyperbolicity for this kind of surfaces.

Gromov Hyperbolicity of Periodic Graphs

“…Characterizing hyperbolic graphs is a main problem in the theory of hyperbolicity; since this is a very ambitious goal, a more achievable (yet very difficult) problem is to characterize hyperbolic graphs in particular classes of graphs. The papers [2,4,[7][8][9]11,12,25,27,[30][31][32]37,39] study the hyperbolicity of complement of graphs, chordal graphs, periodic planar graphs, planar graphs, strong product graphs, line graphs, Cartesian product graphs, cubic graphs, short graphs, median graphs, and different generalizations of chordal graphs; however, characterizations of the hyperbolicity in the corresponding classes are obtained only in a few of them. In a previous work, [8], periodic planar graphs were considered.…”

Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces

“…This has stimulated a good number of works on the subject, e.g. [17,28,29,31,35] for negative constant curvature and [30,32] with negative variable curvature.…”

“…(See[32, Theorem 5.5].) Let us consider a complete Riemannian surface S (with or without boundary) with K −k 2 < 0; if S has boundary, we also require that ∂ S is the union of simple closed geodesics.…”

“…(See[32, Lemma 3.2].) Let us consider R 2 = {(θ, r): θ, r ∈ R} with two different metrics given in Fermi coordinates as ds2 …”

Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics

“…The Gromov hyperbolicity of the Poincare´hyperbolic metric is not as well understood, although several intrinsic results were obtained in [12][13][14][15][16][17][18][19][20], particularly in the case of Denjoy domains. The reason for studying Denjoy domains is that, on the one hand, they are a very general type of Riemann surfaces, and, *Corresponding author.…”