The Milnor-Švarc lemma, is, without doubt, regarded as one of the most important statements in geometric group theory. (Edit) One of the corollaries of this lemma says that if a hyperbolic group $G$ acts geometrically on a hyperbolic space $X$, then the Gromov boundaries of $G$ and $X$ turn out to be homeomorphic.
Now, suppose that $(X, d_X)$ is a complete CAT(0)-space, and $G$ is a finitely-generated group equipped with a geometric action on $X$. Let us fix a point $x_0 \in X$ with the trivial stabilizer, and consider a distance on $G$, which is defined as follows: $d(g_1, g_2) = d_X(g_1.x_0, g_2.x_0)$. We need the stabilizer to be trivial so that $d$ becomes a well-defined distance.
Conjecture: Fix a point $x_0 \in X$. The restriction map $r : \partial_B(X, d_X) \rightarrow \partial_B(G, d)$, where $$ r(h)(g) = h(g.x_0) \text{ for any } h \in \partial_B(X, d_X). $$ is a homeomorphism of Busemann boundaries $\partial_B(X, d_X)$ and $\partial_B(G, d)$. Moreover, if we define the Busemann function and the Busemann cocycle as follows: $$ b_X : X \times \partial X \rightarrow \mathbb{R}, \quad b_X(x, \xi) = \lim\limits_{t \rightarrow \infty}(d(x, \xi(t)) - t), $$ $$ c_B : G \times \partial_B(G, d), \quad c_B(g, \xi) := \xi(g^{-1}) = \lim_{x \rightarrow \xi} (d(x, g^{-1}) - d(e, x)), $$ then $$ c_B(g, r(\xi)) = b_X(g^{-1}.x_0, \xi). $$
Keep in mind that $r$ is well-defined because for complete CAT(0)-spaces the Gromov and Busemann boundaries are homeomorhpic.
This statement, if true, looks quite natural and should be well-known, but I failed to find this result in standard geometric group theory textbooks.
Because I haven't found a proof of this fact, I will attempt to prove this fact myself.
For any $y \in X$ define a function $h_y(x) = d_X(x,y) - d_X(y,x_0)$.
Showing that $r$ is surjective is not that difficult, because $\partial_B(X, d_X)$ is sequentially compact. If a sequence $(h_{g_i})_{i \in \mathbb{N}}$, where $h_{g_i}(x) := d(x, g_i.x_0) - d(g_i.x_0, x_0)$, converges to $h$ in $\partial_B(G, d)$, then we can find a subseqeuence $(h_{g_{i_k}.x_0})_{k \in \mathbb{N}}$ which converges in $\partial_B(X, d_X)$, but this limit, restricted to the orbit $Gx_0$, has to be equal to $h$.
To show that $r$ is injective (this is the non-trivial part!), we need to prove the following statement: if $\xi_1, \xi_2$ are non-asymptotical geodesical rays in $\partial(X)$, then $$\lim_{s\rightarrow \infty}(b(\xi_2(s), \xi_1) + s) = \infty.$$ Because these rays are non-asymptotic, we use the fact that the CAT(0)-angle between them is non-trivial, which allows us to use the CAT(0)-law of cosines in a nice way, so that we get
$$ \begin{gathered} \lim_{t \rightarrow \infty} \sqrt{t^2 + s^2 - 2st \cos( \angle_{x_0}(\xi_1, \xi_2) - \varepsilon)} - t \le \lim_{t \rightarrow \infty} d(\xi_1(t), \xi_2(s)) - t \le \\ \le \lim_{t \rightarrow \infty} \sqrt{t^2 + s^2 - 2st \cos( \angle_{x_0}(\xi_1, \xi_2) + \varepsilon)} - t, \end{gathered} $$
for some very small $\varepsilon > 0$ and a big enough $s > 0$. Here I refer to the Propositon II.9.8(1) in Bridson-Haefliger. Keep in mind that these limits can be computed explicitly, and we finish the argument by taking $\lim\limits_{s \rightarrow \infty}$ and applying the squeeze theorem: $$ -s ( \cos( \angle(\xi_1, \xi_2) - \varepsilon) - 1) \le b(\xi_2(s), \xi_1) + s \le -s (\cos( \angle(\xi_1, \xi_2) + \varepsilon) - 1). $$
Suppose that $r$ isn't injective, then there are distinct horofunctions $h_1, h_2$ which coincide on $Gx_0$. However, because the fundamental domain is compact, and horofunctions are 1-Lipschitz, we get $|h_1 - h_2|$ is a uniformly bounded function on $X$. However, $X$ is CAT(0), so we can consider the corresponding rays $\xi_1, \xi_2$. Due to our assumptions, they are non-asymptotic, and we get $$ \sup_{t} |h_2(\xi_1(t)) - h_1(\xi_1(t))| = \sup_{t} |h_2(\xi_1(t)) + t| = \infty, $$ and this yields a contradiction with the uniform boundedness of $|h_1 - h_2|$.
If this "theorem" is true, then we can use it to explicitly recover the Busemann cocycle for any hyperbolic group acting on a hyperbolic space for which we have a nice description of the Busemann function ($\mathbb{H}^n$, for example). Also, we could use this statement as a tremendously inefficient way to check whether a particular non-elementary hyperbolic group $G$ is not CAT(0): consider all left-invariant distances $d$ on $G$, and prove that for any such $d$ the Busemann boundary $\partial_B(G, d)$ is not homeomorphic to its Gromov boundary $\partial G$. Of course, such an example isn't known...
So, here are my questions: is this a known generalization, and are there better applications? I do admit that due to the fact that the Busemann boundary is not a quasi-isometric invariant of a metric space, we don't have as much freedom and flexibility as in the hyperbolic setting. However, maybe we can use a statement like this in a different way?
Edit: Indeed, as indicated in Moishe Kohan's comment, this looks like a standard application of some properties of CAT(0)-angles between geodesic rays. As far as I understand, the argument presented in my answer can be recovered from the proof of Theorem II.4.4 in the W.Ballman's textbook "Lectures on spaces of nonpositive curvature". However, I am still interested in possible applications, if there are any.
