Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ratio of products of $2$ pairwise differences (using an affine coordinate on the Riemann sphere), so in this sense it has degree $2$.


I came across an expression, depending on $4$ distinct points in $H^3$, say $p\_1$, $p\_2$, $p\_3$ and $p\_4$, more precisely depending on ideal points lying on hyperbolic lines joining pairs of points in that configuration, and such that when the $p\_i$ all lie on the sphere at infinity, for $i = 1, \ldots, 4$, then one gets the usual cross-ratio. Interestingly, this extended cross-ratio has degree $3$, not $2$.


Let $p\_1$, $p\_2$, $p\_3$ and $p\_4$ be points in $H^3$ such that
$$ p\_1 \neq p\_2 \neq p\_3 \neq p\_4.$$


Let $t\_{12}$ denote the ideal point on the Riemann sphere which is the limiting point of the geodesic ray starting at $p\_1$ and passing through $p\_2$, and so on. I will use a complex affine coordinate, so that the $t\_{ab}$ are actually complex numbers (possibly $\infty$). Consider
$$ C(p\_1,p\_2,p\_3,p\_4) = \frac{(t\_{12}-t\_{23})(t\_{21}-t\_{34})(t\_{32}-t\_{43})}{(t\_{12}-t\_{21})(t\_{23}-t\_{32})(t\_{34}-t\_{43})}.$$


In the limiting case where all the points $p\_i$ lie on the sphere at infinity, and let us denote them by $z\_1$, $z\_2$, $z\_3$ and $z\_4$ in this case, we then get


$$ C(z\_1,z\_2,z\_3,z\_4) = \frac{(z\_{2}-z\_{3})(z\_{1}-z\_{4})(z\_{2}-z\_{3})}{(z\_{2}-z\_{1})(z\_{3}-z\_{2})(z\_{4}-z\_{3})}.$$


Simplifying one gets
$$ C(z\_1,z\_2,z\_3,z\_4) = - \frac{(z\_{1}-z\_{4})(z\_{2}-z\_{3})}{(z\_{1}-z\_{2})(z\_{3}-z\_{4})},$$
as claimed.


I wonder if it is new, though I doubt it, as these things are quite classical. But I have only seen the cross-ratio defined for $4$ points on the Riemann sphere, but not for $4$ points in the "bulk", meaning in hyperbolic $3$-space.


Since this extended cross-ratio is invariant under the group of orientation preserving hyperbolic isometries, and gets complex conjugated under orientation reversing hyperbolic isometries, I suspect its real part to be expressible in terms of pairwise hyperbolic distances, and its imaginary part maybe to be expressible in terms of hyperbolic volume (and pairwise hyperbolic distances possibly).


I thus pose the problem of expressing this extended cross-ratio in terms of invariants of hyperbolic geometry (hyperbolic pairwise distances, hyperbolic areas/volumes etc.).


Edit 1: it is known that the volume of an ideal hyperbolic tetrahedron is the Bloch-Wigner function $D\_2$ applied to their cross-ratio. Note that "my" (?) extended cross-ratio $C(p\_1,p\_2,p\_3,p\_4)$ depends real analytically on its arguments, provided $p\_1 \neq p\_2 \neq p\_3 \neq p\_4$. So for instance, $D\_2 \circ C$ is a real analytic extension of the signed hyperbolic volume of an ideal tetrahedron. Could it possibly be the signed hyperbolic volume of the tetrahedron with vertices $p\_i$ ($1 \leq i \leq 4$)? Maybe it does not have the right symmetries... But still, isn't it plausible that $D\_2$ of some analogous real analytic extension of the usual cross-ratio, defined in a way similar to how $C$ is defined, may give the hyperbolic volume of the tetrahedron with vertices $p\_i$?



How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes?
The quantity $C(p\_1,p\_2,p\_3,p\_4)$ does not change if $p\_1$ is replaced by any other point on the ray $p\_2p\_1$. It also does not change if $p\_4$ is replaced by any other point on the ray $p\_3p\_4$. Thus this is an invariant of a triple of oriented lines $\ell\_1 = p\_1p\_2, \ell\_2 = p\_2p\_3, \ell\_3 = p\_3p\_4$. It is thus possible to express $C(p\_1,p\_2,p\_3,p\_4)$ in terms of the following four quantities: angle between $\ell\_1$ and $\ell\_2$, angle between $\ell\_2$ and $\ell\_3$, angle between the planes $\ell\_1\ell\_2$ and $\ell\_2\ell\_3$, distance between $p\_2$ and $p\_3$ (i. e. between the intersection points $\ell\_1 \cap \ell\_2$ and $\ell\_2 \cap \ell\_3$).


For computation let us use Poincare half-space model and send $t\_{23}$ to $0$ and $t\_{32}$ to $\infty \in \mathbb{C}\mathrm{P}^1$. One then has
$$C(p\_1,p\_2,p\_3,p\_4) = \frac{t\_{12}(t\_{21}-t\_{34})}{(t\_{21}-t\_{12})(t\_{34}-t\_{43})}.$$
The points $t\_{12}$ and $t\_{21}$ lie on a line through $0$, on the opposite sides from it, the line $\ell\_1$ is a vertical half-circle with diameter $t\_{12}t\_{21}$. Similarly for $t\_{34}$ and $t\_{43}$. The quotient $\frac{t\_{12}}{t\_{21}-t\_{12}}$ equals $\frac{1-\cos\alpha}{2\cos\alpha}$, where $\alpha$ is the angle between $\ell\_1$ and $\ell\_2$. It is more complicated to decypher $\frac{t\_{21}-t\_{34}}{t\_{34}-t\_{43}}$, but in the end one gets some trigonometric functions of the angles between the lines, and a hyperbolic trigonometric function of the distance $p\_2p\_3$, plugged into some rational function. That is, nothing related to the volume of a simplex.


One more remark: $C(p\_1,p\_2,p\_3,p\_4)$ can be rewritten as a product of certain cross-ratios of points $t\_{ij}$. Thus it has a meaning also for arbitrary lines $\ell\_1, \ell\_2, \ell\_3$.


