Super-Apollonian Generalizations

Question context

In Source citation: MR2173929 the authors introduce the super-Apollonian group, which is generated by the Apollonian group and the corresponding dual group. By replacing the $2$'s in the generators of each subgroup with parameters $\lambda$ and $\mu$ (respectively), one obtains a $2$-parameter generalization of the super-Apollonian group, which we denote $\Gamma_{n,\mu,\lambda}$ (here the $n$ refers to the matrices having size $n \times n$).

Dated source record, inspected 2026-09-10. Current status has not been independently reassessed. Source-reported results have not been independently verified.

Status note in the source

Subproblem 1 is resolved. $\Gamma_{n,\mu,\lambda}$ preserves a quadratic form iff $(n-2)\mu \lambda - 2\mu - 2\lambda = 0$. Moreover, this quadratic form is positive definite when $-2 < \lambda < \frac{2}{n-1}$, and it is Lorentzian when $\lambda < -2$ or $\lambda > \frac{2}{n-1}$.