Mathematics Colloquia and Seminars

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The classification of finite subgroups of SO(3) is a classical problem in geometry. In this talk, we will welcome in the much larger class of rational rotation groups and see what sense can be made of such groups. Call a subgroup G of SO(3,Q) primary if it is discrete in some p-adic topology. These groups are essentially free groups, and we’ll consider them well-understood. We entertain the possibility that any subgroup of SO(3,Q) which is not abelian or primary might be forced to be arithmetic, meaning that it looks a lot like SO(3,A) for a subring A of Q. We prove some results supporting this conjecture, including some special cases of the conjecture and a rigidity result. This conjecture has many analogues in the broader study of discrete subgroups of Lie groups, and has many consequences in geometry and group theory.