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I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:

  1. Higher dimensions
  2. Poincaré disk with radius $r$ and conformal factor $\lambda^r_x=1/(1-\Vert x\Vert^2/r^2)$
  3. A cartesian product of several poincaré disks/balls

The third point is the most important.

Intuitively, I think 1) $\delta$ should be independent of the dimension, 2) increasing the radius $r$ should increase $\delta$ because it makes the space look more Euclidean, and 3) I have no idea.

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    $\begingroup$ A cartesian product will contain an isometrically embedded Euclidean plane. What do you think this should imply? $\endgroup$
    – YCor
    Commented Sep 10, 2018 at 14:22
  • $\begingroup$ Should this imply that a cartesian product of 2 2D poincaré disks is not $\delta$-hyperbolic for any $\delta$? $\endgroup$
    – tisydi
    Commented Sep 10, 2018 at 14:26
  • $\begingroup$ Hi tisydi, welcome to MO. I think I am getting tripped up with your question. Perhaps can sharpen it a little? My confusion stems from the fact, that the Cartesian product of 2 2D Poincare disks is not $\delta$-hyperbolic for any $\delta$ for the comment given above. These questions are good places to start thinking about $\delta$-hyperbolic space, but I fear they do not rise to the research level questions. $\endgroup$
    – Neil Hoffman
    Commented Sep 10, 2018 at 16:55
  • $\begingroup$ Happily, MathStackExchange is the right forum for asking questions. You might peruse some of those questions to see if you can find a satisfying answer there: math.stackexchange.com/search?q=delta+hyperbolic as your question stands now, I am voting to migrate to MSE, but I would encourage you to post to MO in the future. $\endgroup$
    – Neil Hoffman
    Commented Sep 10, 2018 at 16:55

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