Orbits of some special actions on solution set of a arithmetic equation Let $g_1(x,y,z)=(y,x,-z), g_2(x,y,z)=(y,x+y+2z,-y-z)$,
$V= \{(x,y,z)\in Z^3|xy-z^2+1=0 \}$.
Is it possible to find all orbits of the action of group $\langle g_1 \rangle * \langle g_2 \rangle$ on $V$?  How can I find all of them? 
 A: Have you seen this note of Don Zagier?


*

*A One-Sentence Proof that every $p \equiv 1 \mod 4$ is a sum of two squares?


His map might be related:
$$ (x,y,z) \mapsto \left\{ 
\begin{array}{cl} (x+2z, z, y - x - z ) & \text{if }x< y - z \\
(2y-x, y, x - y + z) & \text{if }y-z < x < 2y \\
(x-2y, x-y+z,y)& \text{if }x > 2y
 \end{array}
 \right. $$
He claims (but does not really prove) it is involution on  $X = \{ x^2 + 4yz = p\}$ with exactly one fixed point - if $p$ is prime - so that involution $(x,y,z) \mapsto (y,x,z)$ is also a fixed point.

It is simplification of this argument by Roger Heath-Brown.
A: Not an answer, but some comments:


*

*The quadratic form $q$ (as in YCor's comment) has signature $(-, -, +)$ so the group is a fuchsian group.

*A fuchsian group is Zariski-dense if and only if it is non-elementary. It is pretty clear that this group is non-elementary (because $g_1$ acts non-trivially on the limit points on $g_2.$)

*One does not even need the fact above, but using the results of T. Weigel, one can check surjectivity (onto $SO(3, p))$ for a single $p\geq 3$.

*So, as Yves says, there is one orbit mod large $p$, but

*This does not give us much, since the most studied of such objects - the Apollonian group (see the original papers of Grahah, Lagarias, Mallows, Wilks, but also the amazing later papers of Sarnak, Fuchs, Bourgain, Gamburd...) is likewise Zariski dense, but, nevertheless, has infinitely many orbits.


Finally, it should be remarked that your $V$ is the de Sitter sphere (sphere of radius $1$ for your quadratic form). That makes life slightly more annoying than if it were the hyperbolic plane.
