A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a horosphere.

My question is: does a similar exponential decay still holds in a Gromov hyperbolic space ?

The precise statement would be something like: Let X be a $\delta$-hyperbolic space, $B$ a horoball and $H$ the corresponding horosphere. Assume that nearest point projection on $H$ is well defined and denote by $\pi$ this projection.

There exists $\alpha>0$ such that the following holds. Assume that $\gamma$ is a path in $X\setminus B$ such that $d(\gamma(t),H)\geq k>0$ for some $\alpha$ and for every $t$. Then, $l(\pi \circ \gamma)\leq e^{-\alpha k}l(\gamma)$, where $l$ is the length of a path.

Is there a reference for such a statement or something similar?

EDIT: Maybe the assumption that nearest point projection is too restrictive. A slightly weaker (but more general) statement would be:

Without assuming nearest point projection is well-defined but with the same notations, there exists $\alpha>0$ such that the following holds. Let $x,y\in X$ be such that there exists a path $\gamma$ from $x$ to $y$ such that for all $t$, $d(\gamma(t),H)\geq k>0$. Assume that $x$ and $y$ (coarsely) project onto $p$ and $q$ on the horosphere $H$. Then, $d(x,y)\geq e^{\alpha k}d(p,q)$.

(Maybe you should also require that $p$ and $q$ are far enough from each other, i.e. $d(p,q)\geq a$, where $a$ only depends on the hyperbolic parameters of $X$).

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    $\begingroup$ Your first statement refers to a question in a paper? in a talk? in MathOF?... $\endgroup$ – YCor Jul 5 '17 at 15:40
  • $\begingroup$ Sorry. The first statement refers to a question on MathOF: link $\endgroup$ – M. Dus Jul 5 '17 at 15:45
  • $\begingroup$ What does "projection" mean in a delta hyperbolic space? $\endgroup$ – David Cohen Jul 5 '17 at 16:02
  • $\begingroup$ You are absolutely right. In general nearest point projection is not defined. If the space is proper you can coarsely define it, but the projection won't be unique. Actually, in my situation, I have a nearest point projection. I edited my question. Thank you anyway. $\endgroup$ – M. Dus Jul 6 '17 at 7:26
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    $\begingroup$ Even the claim under the assumption of existence of a Lipschitz nearest point projection is false. Just add a flat strip to the horoball in the hyperbolic plane (along the horocycle). $\endgroup$ – Misha Jul 11 '17 at 16:52

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