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While working on my recent paper with Ben Green, I was introduced to the beautiful theorems of Marina Ratner on unipotent flows on homogeneous spaces, and their application to questions in number theory, such as the Oppenheim conjecture (first solved by Margulis, by establishing what can retrospectively be viewed as a special case of Ratner’s theorems). This is a subject that I am still only just beginning to learn, but hope to understand better in the future, especially given that quantitative analogues of Ratner’s theorems should exist, and should have even more applications to number theory (see for instance this recent paper of Einsiedler, Margulis, and Venkatesh). In this post, I will try to describe some of the background for this theorem and its connection with the Oppenheim conjecture; I will not discuss the proof at all, largely because I have not fully understood it myself yet. For a nice introduction to these issues, I recommend Dave Morris’ recent book on the subject (and this post here is drawn in large part from that book).

Ratner’s theorem takes place on a homogeneous space. Informally, a homogeneous space is a space X which looks “the same” when viewed from any point on that space. For instance, a sphere S^2 is a homogeneous space, but the surface of a cube is not (the cube looks different when viewed from a corner than from a point on an edge or on a face). More formally, a homogeneous space is a space X equipped with an action (g,x) \mapsto gx of a group G of symmetries which is transitive: given any two points x, y on the space, there is at least one symmetry g that moves x to y, thus y=gx. (For instance the cube has several symmetries, but not enough to be transitive; in contrast, the sphere S^2 has the transitive action of the special orthogonal group SO(3) as its symmetry group.) It is not hard to see that a homogeneous space X can always be identified (as a set with an action of G) with a quotient G/\Gamma := \{ g\Gamma: g \in G \}, where \Gamma is a subgroup of G; indeed, one can take \Gamma to be the stabiliser \Gamma := \{ g \in G: gx = x \} of an arbitrarily chosen point x in X, and then identify g\Gamma with g\Gamma x = gx. For instance, the sphere S^2 has an obvious action of the special orthogonal group SO(3), and the stabiliser of (say) the north pole can be identified with SO(2), so that the sphere can be identified with SO(3)/SO(2). More generally, any Riemannian manifold of constant curvature is a homogeneous space; for instance, an m-dimensional torus can be identified with {\Bbb R}^m/{\Bbb Z}^m, while a surface X of constant negative curvature can be identified with SL(2,{\Bbb R})/\Gamma for some subgroup \Gamma of SL(2,{\Bbb R}) (e.g. the hyperbolic plane {\Bbb H} is isomorphic to SL(2,{\Bbb R})/SO(2)). Furthermore, the cosphere bundle S^* X of X – the space of unit (co)tangent vectors on X – is also a homogeneous space with structure group SL(2,{\Bbb R}). (For instance, the cosphere bundle S^* {\Bbb H} of the hyperbolic plane {\Bbb H} is isomorphic to SL(2,{\Bbb R}) / \{ +1, -1 \}.)

For the purposes of Ratner’s theorem, we only consider homogeneous spaces X in which the symmetry group G is a connected finite-dimensional Lie group, and X is finite volume (or more precisely, it has a finite non-trivial G-invariant measure). Every compact homogeneous space is finite volume, but not conversely; for instance the modular curve SL(2,{\Bbb R})/SL(2,{\Bbb Z}) is finite volume but not compact (it has a cusp). (The modular curve has two real dimensions, but just one complex dimension, hence the term “curve”; rather confusingly, it is also referred to as the “modular surface”. As for the term “modular”, observe that the moduli space of unimodular lattices in {\Bbb R}^2 has an obvious action of SL(2,{\Bbb R}), with the stabiliser of {\Bbb Z}^2 being SL(2,{\Bbb Z}), and so this moduli space can be identified with the modular curve.)

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