It is the space $\mathbb{M}^4$, that is, the manifold $\mathbb{R}^4$ with coordinates $(x,y,z,t)$ but with the pseudo-Riemannian metric given in each tangent space by
$$ ds^2=dx^2+dy^2+dz^2-dt^2. $$This space with this pseudo-metric is called Minkowski space, and can be also denoted by $\mathbb{R}^{3,1}$.
The role of spheres in $\mathbb{E}^4$ (points at a fixed distance $d$ from the origin) is played here by hyperboloids. So the group $O(4)$ is replaced by the Lorentz group $O(3,1)$, which leaves the pseudo-metric invariant, and the hyperboloids. It also leaves invariant the cone
$$ t^2-x^2-y^2-z^2=0 $$The Minkowski space is a homogeneous space $G/H$ where $G$ is the Poincare group and $H$ is the Lorentz group.
Some isometries of $\mathbb{M}^4$ are called hyperbolic rotations, but I don't know if has to do with isometries of the hyperbolic plane (but see the following section). In some sense the idea is that if we consider rotations in $x,y,z$ we have usual spatial rotations (the "3" in $O(3,1)$)). But if we consider rotations with $t$ and other coordinate we have these "weird" hyperbolic rotations called Lorentz boost.
In the note Elliptic, hyperbolic and parabolic geometry it is shown that the three constant curvature models (including hyperbolic geometry) can be achieved within $\mathbb{M}^4$.
In this link there is a good answer to this question. It gives a similar idea than in Elliptic, hyperbolic and parabolic geometry, but includes a idea of the stereographic projection of the hyperboloid, which is very interesting.
When we make this stereographic projection we obtain the Poincare disk. And I guess that the elements of $O(3,1)$ acting on $\mathbb{M}^4$ yields the isometries of $\mathbb{D}$. So, in particular, hyperbolic rotations in the sense of Special Relativity would correspond to a kind of isometries of the hyperbolic plane.
I have to think if this has to do with what is said at Moebius transformations#Relation to Special Relativity. Basically: Moebius transformations are the Lorentz transformations.
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Author of the notes: Antonio J. Pan-Collantes
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