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Two easy questions:

Given a metric space $(Z,d_Z)$, let $X = \text{Con}_h(Z) = Z \times ]0,\infty[$ and for $(z,i), (z',i') \in \text{Con}_h(Z)$ define a metric on $X$ as follows: $d((z,i), (z',i')) = d_{\mathbb{H}^2}((0,i), (d_Z(z,z'),i') = \text{arcosh}(1+\frac{d_Z(z,z')^2 + (i-i')^2}{2ii'})$.

Question: If $Z$ was a $\text{CAT}(0)$ space, is then $X$ $\text{CAT}(-1)$?

Question 2: If not, is there a way to construct a $\text{CAT}(-1)$ space that has a given $\text{CAT}(0)$ space as boundary?

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    $\begingroup$ Did you look at the relevant section of Bridson and Haefliger? What you have in mind is a limiting case of their construction and, hence, will be CAT(-1). But its ideal boundary will be larger than $Z$. $\endgroup$
    – Misha
    Commented Sep 13, 2017 at 19:57

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