An asymptotic for Markoff-Hurwitz tuples

We establish an asymptotic formula for the number of integer solutions
to the Markoff-Hurwitz equation
\[
x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k.
\]
When $n\geq4$ the previous best result is by Baragar (1998)
that gives an exponential rate of growth with exponent $\beta$ that
is not in general an integer. We give a new interpretation
of this exponent of growth in terms of the unique parameter for which
there exists a certain conformal measure on projective space. In a
special case, in light of recent results of Huang and Norbury our counting result
can be rephrased in terms of the simple closed curves on a special
arithmetic 3 times punctured projective plane.