The Milnor-Švarc lemma, is, without doubt, regarded as one of the most important statements in geometric group theory. Roughly, it states 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) = limlimits_{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 $$lim_{srightarrow infty}(b(xi_2(s), xi_1) + s) = infty.$$ 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 $limlimits_{s rightarrow infty}$ 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?**