Fundamental domain for the Markoff-Hurwitz equation

Abstract

For integers $a\neq0$, $k$, and $n\geq3$, we consider the Markoff-Hurwitz equation given by $x_1^1+\cdots+x_n^2-ax_1\cdots x_n=k$. By defining graphs associated with a height function and by using their properties, we find an exact fundamental domain for a symmetric group generated by involution maps sending $(x_1,\dots,x_n)$ to $(x_1,\dots,ax_1\cdots x_{i-1}x_{i+1}\cdots x_n-x_i,\dots,x_n)$, permutations, and double sign changes on the set of integral solutions for the Markoff-Hurwitz equation.