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1. Vinberg’s Algorithm

In this section, $\Gamma < \operatorname{O}(n,1)$ is taken to be an arithmetic lattice. We’ll write $\Gamma = R \rtimes \operatorname{Aut}(P)$ where $R$ is generated by reflections and $\operatorname{Aut}(P)$ is the group of symmetries of the Coxeter polytope $P$ corresponding to $R$.

    1. Decidability

          Vinberg’s algorithm terminates if $\Gamma$ is reflective (i.e. if $|\operatorname{Aut}(P)| < \infty$), and does not terminate if $\Gamma$ is not reflective. As such, determining whether such a given arithmetic lattice is reflective is at least semi-decidable.

      Add remark

      Problem 1.1.

      [Michael Kapovich] Is reflectivity of a maximal arithmetic lattice a decidable problem? If so, can we find an efficient algorithm to solve this decision problem?
        1. Remark. [Michael Kapovich] Yes, the problem is decidable. One way to proceed is to search for infinite order automorphisms of the fundamental domain $P$ of the maximal reflection subgroup $R$ in the lattice $\Gamma$. The way it works is that on the step $k$ of the inductive construction of $P$ we have a polyhedron $P_k$ (the polyhedra are nested: $P_1\supset P_2\supset ... \supset P_k \supset ... \supset P$) we search for a pair of vertices $u, v$ of $P_k$ (possibly ideal or even hyperfinite) and an isometry $\gamma\in\Gamma$ sending the link of $u$ in $P_k$ to the link of $v$ in $P_k$. If there exists such non-elliptic $\gamma$, then $R$ has infinite index in $\Gamma$ and vice-versa.


            • Add remark

              Problem 1.2.

              Suppose $\Gamma$ is reflective. Given a bound for the volume of the fundamental domain, $D_\Gamma$, can we also find a bound on the number of walls in $D_\Gamma$?


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                  Problem 1.3.

                  If $\Gamma$ is non-reflective, then $\operatorname{Aut}(P)$ contains an infinite-order element. How many walls of the fundamental domain to we need to find via Vinberg’s algorithm in order to detect this infinite-order symmetry?


                    • Add remark

                      Problem 1.4.

                      Suppose $\Gamma$ is non-reflective. After running Vinberg’s algorithm to find part of the Coxeter polytope $P$, we can look for symmetries in the Coxeter diagram and deduce from these some $\operatorname{O}(q)$ symmetries of the polytope. Can we use these symmetries to find new roots and possibly the infinite-order symmetry?

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