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3. Quasi-reflective and Quasi-arithmetic groups

    1. Quasi-reflective Groups

          A maximal arithmetic lattice $\Gamma = R \rtimes \operatorname{Aut}(P)$ is quasi-reflective if $\operatorname{Aut}(P)$ is infinite and virtual abelian.

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      Problem 3.1.

      There are only finitely-many quasi-reflective arithmetic lattices in each dimension, and we have a complete list in dimension $n=3$. Can we classify all of them for $n > 3$?


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          Problem 3.2.

          Do these quasi-reflective groups contain any salient features that might suggest an effective algorithm for detecting infinite-order symmetries in general (and hence an effective halting condition for Vinberg’s algorithm)?

            • Quasi-arithmetic Groups


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              Problem 3.3.

              To what extent are quasi-arithmetic groups like arithmetic reflection groups? Are there only finitely many maximal quasi-arithmetic reflective lattices?

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