Geometry and Groups: Enumeration and Finite Representations

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The richest symmetry groups in nature are exemplified by the collection of all rotations of a two-dimensional sphere or the collection of all rigid motions of the Euclidean plan. The operations of applying two symmetries in order, and of inverting a symmetry to get another one, endow these sets of symmetries with the structure known in mathematics as a group. Many other groups arise as subgroups of these groups of continuous motions, modeled on the group of rigid motions of a planar tiling and its relationship to the full group of isometries of the Euclidean plan. Number theory give rise to many of these examples, with constructions that are analogous to the discrete, widely separated way that the integers sit within the real number line. One of the projects to be pursued seeks to find a computationally feasible way to list all of the arithmetically defined discrete subgroups of motion within the most important groups of continuous motions.The continuous groups of motion referred to above are known in mathematics as semisimple Lie groups; Sophus Lie was the nineteenth-century mathematician who discovered the basic structural facts of such groups of continuous transformations, and semisimplicity is an algebraic property shared by most of the important examples. The first line of work planned will recursively enumerate arithmetic lattices in semisimple Lie groups, providing a solution to the isomorphism problem for this class of groups and addressing a conjecture of Belolipetsky and Lubotzky on the number of isometries between distinct finite covers of an arithmetic manifold. A second line of investigation concerns the relationship between a manifold and its finite-sheeted covering spaces, which is encoded in the profinite completion of the manifold's fundamental group.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.