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4. Apollonian-Type Groups

    1. Generalizations


      Add remark

      Problem 4.1.

      What is the appropriate $n$-dimensional generalization of the Apollonian packing?

        • Super-Apollonian Generalizations

              In [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 $n \times n$).

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          Problem 4.2.

          [Jeffrey Lagarias] For which parameter values...
          1. does $\Gamma(\mu,\lambda)$ preserve a quadratic form $q$?
          2. is $\Gamma(\mu,\lambda)$ discrete?
          3. is $\Gamma(\mu,\lambda)$ a lattice?
          4. is $\Gamma(\mu,\lambda)$ (quasi-)arithmetic?
          5. does $(\mu,\lambda)$ have extra relations not already present in the Apollonian/dual Apollonian subgroups?
              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 $\lambda > \frac{2}{n-1}$.

              Cite this as: AimPL: Arithmetic reflection groups and crystallographic packings, available at http://aimpl.org/reflectiongpv.