5 clarify a point

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:

• Riemann Sphere $\mathbb{C}\cup{\infty}$ (elliptic type).
• Complex plane (parabolic type).
• Open unit disk (hyperbolic type).

On the other hand, given a Riemann surface one can asks if it is hyperbolic in the Gromov's sense. In other words, does there exists $\delta>0$ such that all the geodesic triangles in the surface are $\delta$-thin?

It seems to me that this two notions of hyperbolicity are not equivalent and one can have counterexamples in both directions. For instance, the two dimensional torus $\mathbb{T}^2$ is hyperbolic in Gromov's sense (since it is compact) compact), but it is conformally equivalent to it's also a quotient of the complex Euclidean plane by a free action of a discrete group of isometries and therefore, of parabolic type.

My questions are: what is a sufficient condition to guarantee that for a surface of hyperbolic type is also to be Gromov's hyperbolic? what is known about the relation of these two notions?

Related Question: Let $G$ be an infinite planar graph with uniformly bounded degree and assume that the simple random walk is transient. Is the graph necessarily Gromov's hyperbolic?

4 imprved the english

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:

• Riemann Sphere $\mathbb{C}\cup{\infty}$ (elliptic type).
• Complex plane (parabolic type).
• Open unit disk (hyperbolic type).

On the other hand, given a Riemann surface one can asks if it is hyperbolic in the Gromov's sense. In other words, does there exists $\delta>0$ such that all the geodesic triangles in the surface are $\delta$-thin?

It seems to me that this two notions of hyperbolicity are not equivalent and one can have counterexamples in both directions. For instance, the two dimensional torus $\mathbb{T}^2$ is hyperbolic in Gromov's sense (since it is compact) but it is conformally equivalent to the complex plane and therefore of parabolic type.

My questions are: what is a necessary sufficient condition to guarantee that a surface of hyperbolic type is also Gromov's hyperbolic? what is known about the relation of these two notions?

Related Question: Let $G$ be an infinite planar graph with uniformly bounded degree and assume that the simple random walk is transient. Is the graph necessarily Gromov's hyperbolic?

For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a simply connected Riemann surfaces is conformally equivalent to one of the following:

• Riemann Sphere $\mathbb{C}\cup{\infty}$ (elliptic type).
• Complex plane (parabolic type).
• Open unit disk (hyperbolic type).

On the other hand, given a Riemann surface one can asks if it is hyperbolic in the Gromov's sense. In other words, does there exists $\delta>0$ such that all the geodesic triangles in the surface are $\delta$-thin?

It seems to me that this two notions of hyperbolicity are not equivalent and one can have counterexamples in both directions. For instance, the two dimensional torus $\mathbb{T}^2$ is hyperbolic in Gromov's sense (since it is compact) but it is conformally equivalent to the complex plane and therefore of parabolic type.

My questions are: what is a necessary condition to guarantee that a surface of hyperbolic type is also Gromov's hyperbolic? what is known about the relation of these two notions?

Related Question: Let $G$ be an infinite planar graph with uniformly bounded degree and assume that the simple random walk is transient. Is the graph necessarily Gromov's hyperbolic?

2 fixed some typos
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