An asymptotic for the growth of Markoff-Hurwitz tuples

For integer parameters $n \geq 3$, $a \geq 1$, and $k \geq 0$ the Markoff-Hurwitz equation is the diophantine equation \[ x_1^2 + x_2^2 + \cdots + x_n^2 = ax_1x_2 \cdots x_n + k.\] In this talk, we establish an asymptotic count for the number of integral solutions with $\max \{ x_1, x_2, \dots, x_n \} \leq R$. When $n = a = 3$ and $k=0$ this equation is known simply as the Markoff equation, for which the asymptotic count was studied in detail by Zagier in 1982. The previous best result for $n \geq 4$ is due to Baragar in 1998 who established an exponential rate of growth with exponent $\beta(n) > 0$ when $k=0$, and which is not, in general, an integer. We use methods from symbolic dynamics to improve this asymptotic count, and which yield a new interpretation of this exponent $\beta$ as the unique parameter for which there exists a certain conformal measure on projective space. Joint work with Alex Gamburd and Michael Magee.