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4. Apollonian-Type Groups

    1. Generalizations


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      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.