The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of , which is simpler in a different way (due to the semisimplicity of , and lack of compact factors).
— Nilpotent groups –
Before we can get to Ratner-type theorems for nilmanifolds, we will need to set up some basic theory for these nilmanifolds. We begin with a quick review of the concept of a nilpotent group – a generalisation of that of an abelian group. Our discussion here will be purely algebraic (no manifolds, topology, or dynamics will appear at this stage).
Definition 1. (Commutators) Let G be a (multiplicative) group. For any two elements g,h in G, we define the commutator [g,h] to be (thus g and h commute if and only if the commutator is trivial). If H and K are subgroups of G, we define the commutator to be the group generated by all the commutators .
For future reference we record some trivial identities regarding commutators:
Exercise 1. Let H, K be subgroups of a group G.
- Show that [H,K] = [K,H].
- Show that H is abelian if and only if [H,H] is trivial.
- Show that H is central if and only if [H,G] is trivial.
- Show that H is normal if and only if .
- Show that [H,G] is always normal.
- If is a normal subgroup of both H and K, show that .
- Let HK be the group generated by . Show that is a normal subgroup of HK, and when one quotients by this subgroup, the images of H and K are groups that commute with each other.
Exercise 2. Let G be a group. Show that the group G/[G,G] is abelian, and is the universal abelianisation of G in the sense that every homomorphism from G to an abelian group H can be uniquely factored as , where is the quotient map and is a homomorphism.
Definition 2. (Nilpotency) Given any group G, define the lower central series
by setting and for . We say that G is nilpotent of step s if is trivial (and is non-trivial).
Examples 1. A group is nilpotent of step 0 if and only if it is trivial. It is nilpotent of step 1 if and only if it is non-trivial and abelian. Any subgroup or homomorphic image of a nilpotent group of step s is nilpotent of step at most s. The direct product of two nilpotent groups is again nilpotent, but the semi-direct product of nilpotent groups is merely solvable in general. If G is any group, then is nilpotent of step at most s.
Example 2. Let be an integer, and let
be the group of all upper-triangular real matrices with s on the diagonal (i.e. the group of unipotent upper-triangular matrices). Then is nilpotent of step n. Similarly if is replaced by other fields.
Exercise 3. Let G be an arbitrary group.
- Show that each element of the lower central series is a characteristic subgroup of G, i.e. for all automorphisms . (Specialising to inner automorphisms, this shows that the are all normal subgroups of G.)
- Show the filtration property for all . (Hint: induct on i+j; then, holding i+j fixed, quotient by , and induct on (say) i. Note that once one quotients by , all elements of are central (by the first induction hypothesis), while commutes with (by the second induction hypothesis). Use these facts to show that all the generators of commute with .)
Exercise 4. Let G be a nilpotent group of step 2. Establish the identity
for any integer n and any , where . (This can be viewed as a discrete version of the first two terms of the Baker-Campbell-Hausdorff formula.) Conclude in particular that the space of Hall-Petresco sequences , where for , is a group under pointwise multiplication (this group is known as the Hall-Petresco group of G). There is an analogous identity (and an analogous group) for nilpotent groups of higher step; see for instance this paper of Leibman for details. The Hall-Petresco group is rather useful for understanding multiple recurrence and polynomial behaviour in nilmanifolds; we will not discuss this in detail, but see Exercise 5 below for a hint as to the connection.
Exercise 5. (Arithmetic progressions in nilspaces are constrained) Let X be a nilspace of step , and consider two arithmetic progressions and of length s+2 in X, where and . Show that if these progressions agree in the first s+1 places (thus for all then they also agree in the last place. (Hint: the only tricky case is s=2. For this, either use direct algebraic computation, or experiment with the group of Hall-Petresco sequences from the previous exercise. The claim is in fact true for general s, because the Hall-Petresco group exists in every step.)
Remark 1. By Exercise 3.2, the lower central series is a filtration with respect to the commutator operation . Conversely, if G admits a filtration with and trivial for , then it is nilpotent of step at most s. It is sometimes convenient for inductive purposes to work with filtrations rather than the lower central series (which is the “minimal” filtration available to a group G); see for instance my paper with Ben Green on this topic.
Remark 2. Let G be a nilpotent group of step s. Then is trivial and so is central (by Exercise 1), thus abelian and normal. By another application of Exercise 1, we see that is nilpotent of step s-1. Thus we see that any nilpotent group G of step s is an abelian extension of a nilpotent group of step s-1, in the sense that we have a short exact sequence
where the kernel is abelian. Conversely, every abelian extension of an s-1-step nilpotent group is nilpotent of step at most s. In principle, this gives a recursive description of s-step nilpotent groups as an s-fold iterated tower of abelian extensions of the trivial group. Unfortunately, while abelian groups are of course very well understood, abelian extensions are a little inconvenient to work with algebraically; the sequence (6) is not quite enough, for instance, to assert that G is a semi-direct product of and (this would require some means of embedding back into G, which is not available in general). One can identify G (using the axiom of choice) with a product set with a group law , where is a map obeying various cocycle-type identities, but the algebraic structure of is not particularly easy to exploit. Nevertheless, this recursive tower of extensions seems to be well suited for understanding the dynamical structure of nilpotent groups and their quotients, as opposed to their algebraic structure (cf. our use of recursive towers of extensions in our previous lectures in dynamical systems and ergodic theory).
In our applications we will not be working with nilpotent groups G directly, but rather with their homogeneous spaces X, i.e. spaces with a transitive left-action of G. (Later we will also add some topological structure to these objects, but let us work in a purely algebraic setting for now.) Such spaces can be identified with group quotients where is the stabiliser of some point x in X. (By the transitivity of the action, all stabilisers are conjugate to each other.) It is important to note that in general, is not normal, and so X is not a group; it has a left-action of G but not right-action of G. Note though that any central subgroup of G acts on either the left or the right.
Now let G be s-step nilpotent, and let us temporarily refer to as an s-step nilspace. Then G_s acts on the right in a manner that commutes with the left-action of G. If we set , we see that the right-action of on is trivial; thus we in fact have a right-action of the abelian group . (In our applications, will be a torus.) This action can be easily verified to be free. If we let be the quotient space, then we can view X as a principal -bundle over . It is not hard to see (cf. the isomorphism theorems) that , where is the quotient map. Observe that is nilpotent of step s-1, and is a subgroup. Thus we have expressed an arbitrary s-step nilspace as a principal bundle (by some abelian group) over an s-1-step nilspace, and so s-step nilspaces can be viewed as towers of abelian principal bundles, just as s-step nilpotent groups can be viewed as towers of abelian extensions.
– Nilmanifolds –
It is now time to put some topological structure (and in particular, Lie structure) on our nilpotent groups and nilspaces.
Definition 3 (Nilmanifolds). An s-step nilmanifold is a nilspace , where G is a finite-dimensional Lie group which is nilpotent of step s, and is a discrete subgroup which is cocompact or uniform in the sense that the quotient is compact.
Remark 3. In the literature, it is sometimes assumed that the nilmanifold is connected, and that the group G is connected, or at least that its group of connected components ( being the identity component of ) is finitely generated (one can often easily reduce to this case in applications). It is also convenient to assume that is simply connected (again, one can usually reduce to this case in applications, by passing to the universal cover of if necessary), as this implies (by the Baker-Campbell-Hausdorff formula) that the nilpotent Lie group is exponential, i.e. the exponential map is a homeomorphism.
Example 3. (Skew torus) If we define
(thus G consists of the upper-triangular unipotent matrices whose middle right entry is an integer, and is the subgroup in which all entries are integers) then is a 2-step nilmanifold. If we write
then we see that is isomorphic to the square with the identifications and . (Topologically, this is homeomorphic to the ordinary 2-torus , but the skewness will manifest itself when we do dynamics.)
Example 4. (Heisenberg nilmanifold) If we set
then is a 2-step nilmanifold. It can be viewed as a three-dimensional cube with the faces identified in a somewhat skew fashion, similarly to the skew torus in Example 3.
Let be the Lie algebra of G. Every element g of G acts linearly on by conjugation. Since G is nilpotent, it is not hard to see (by considering the iterated commutators of g with an infinitesimal perturbation of the identity) that this linear action is unipotent, and in particular has determinant 1. Thus, any constant volume form on this Lie algebra will be preserved by conjugation, which by basic differential geometry allows us to create a volume form (and hence a measure) on G which is invariant under both left and right translation; this Haar measure is clearly unique up to scalar multiplication. (In other words, nilpotent Lie groups are unimodular.) Restricting this measure to a fundamental domain of and then descending to the nilmanifold we obtain a left-invariant Haar measure, which (by compactness) we can normalise to be a Borel probability measure. (Because of the existence of a left-invariant probability measure on , we refer to the discrete subgroup of G as a lattice.) One can show that this left-invariant Borel probability measure is unique.
Definition 4. (Nilsystem) An s-step nilsystem (or nilflow) is a topological measure-preserving system (i.e. both a topological dynamical system and a measure-preserving system) with underlying space a s-step nilmanifold (with the Borel -algebra and left-invariant probability measure), with a shift T of the form for some .
Example 5. The Kronecker systems on compact abelian Lie groups are 1-step nilsystems.
Example 6. The skew shift system on the torus can be identified with a nilflow on the skew torus (Example 3), after identifying (x,y) with [x,y] and using the group element
to create the flow.
Example 7. Consider the Heisenberg nilmanifold (Example 4) with a flow generated by a group element
for some real numbers . If we identify
then one can verify that
where and are the integer part and fractional part functions respectively. Thus we see that orbits in this nilsystem are vaguely quadratic in n, but for the presence of the not-quite-linear operators and . (These expressions are known as bracket polynomials, and are intimately related to the theory of nilsystems.)
Given that we have already seen that nilspaces of step s are principal abelian bundles of nilspaces of step s-1, it should be unsurprising that nilsystems of step s are abelian extensions of nilsystems of step s-1. But in order to ensure that topological structure is preserved correctly, we do need to verify one point:
Lemma 1. Let be an s-step nilmanifold, with G connected and simply connected. Then is a discrete cocompact subgroup of . In particular, is a compact connected abelian Lie group (in other words, it is a torus).
Proof. Recall that G is exponential and thus identifiable with its Lie algebra . The commutators can be similarly identified with the Lie algebra commutators ; in particular, the are all connected, simply connected Lie groups.
The key point to verify is the cocompact nature of in ; all other claims are straightforward. We first work in the abelianisation , which is identifiable with its Lie algebra and thus isomorphic to a vector space. The image of under the quotient map is a cocompact subgroup of this vector space; in particular, it contains a basis of this space. This implies that contains an “abelianised” basis of G in the sense that every element of G can be expressed in the form modulo an element of the normal subgroup for some real numbers , where we take advantage of the exponential nature of G to define real exponentiation . Taking commutators s times (which eliminates all the “modulo ” errors), we then see that is generated by expressions of the form for and real t. Observe that these expressions lie in if t is integer. As is abelian, we conclude that each element in can be expressed as an element of , times a bounded number of elements of the form with . From this we conclude that the quotient map is already surjective on some bounded set, which we can take to be compact, and so is compact as required.
As a consequence of this lemma, we see that if is an s-step nilmanifold with G connected and simply connected, then is an s-1-step nilmanifold (with G still connected and simply connected), and that X is a principal -bundle over in the topological sense as well as in the purely algebraic sense. One consequence of this is that every s-step nilsystem (with G connected and simply connected) can be viewed as a toral extension (i.e. a group extension by a torus) of an s-1-step nilsystem (again with G connected and simply connected). Thus for instance the skew shift system (Example 6) is a circle extension of a circle shift, while the Heisenberg nilsystem (Example 7) is a circle extension of an abelian 2-torus shift.
Remark 4. One should caution though that the converse of the above statement is not necessarily true; an extension of an s-1-step nilsystem X by a torus T using a cocycle need not be isomorphic to an s-step nilsystem (the cocycle has to obey an an additional equation (or more precisely, a system of equations when ), known as the Conze-Lesigne equation, before this is the case. See for instance this paper of Ziegler for further discussion.
Exercise 6. Show that Lemma 1 continues to hold if we relax the condition that G is connected and simply connected, to instead require that is connected, that is finitely generated, and that is simply connected. (I believe that all three of these hypotheses are necessary, but haven’t checked this carefully.)
Exercise 7. Show that Lemma 1 continues to hold if and are replaced by and for any . In particular, setting i=2, we obtain a projection map from X to the Kronecker nilmanifold .
Remark 5. One can take the structural theory of nilmanifolds much further, in particular developing the theory of Mal’cev bases (of which the elements used to prove Lemma 1 were a very crude prototype). See the foundational paper of Mal’cev (or its English translation) for details, as well as the later paper of Leibman which addresses the case in which G is not necessarily connected.
– A criterion for ergodicity –
We now give a useful criterion to determine when a given nilsystem is ergodic.
Theorem 1. Let be an s-step nilsystem with G connected and simply connected, and let be the underlying Kronecker factor, as defined in Exercise 7. Then X is ergodic if and only if is ergodic.
This result is originally due to Leon Green, using spectral theory methods. We will use an argument of Parry (and adapted by Leibman), relying on “vertical” Fourier analysis and topological arguments, which we have already used for the skew shift in Proposition 1 of Lecture 9.
Proof. If X is ergodic, then the factor is certainly ergodic. To prove the converse implication, we induct on s. The case is trivial, so suppose and the claim has already been proven for s-1. Then if is ergodic, we already know from induction hypothesis that is ergodic. Suppose for contradiction that X is not ergodic, then we can find a non-constant shift-invariant function on X. Using Fourier analysis (or representation theory) of the vertical torus as in Proposition 1 of Lecture 9, we may thus find a non-constant shift-invariant function f which has a single vertical frequency in the sense that one has for all , , and some character . If the character is trivial, then f descends to a non-constant shift-invariant function on , contradicting the ergodicity there, so we may assume that is non-trivial. Also, |f| descends to a shift-invariant function on and is thus constant by ergodicity; by normalising we may assume .
Now let , and consider the function . As is central, we see that is -invariant and thus descends to . Furthermore, as f is shift-invariant (so ), and , some computation reveals that is an eigenfunction:
In particular, if , then must have mean zero. On the other hand, by continuity (and the fact that |f|=1) we know that has non-zero mean for close enough to the identity. We conclude that for all close to the identity; as the map is a homomorphism, we conclude in fact that for all . In particular, from (14) and ergodicity we see that is constant, and so for some .
Now let be arbitrary. Observe that
Remark 6. The hypothesis that G is connected and simply connected can be dropped; see the paper of Leibman for details.
One pleasant fact about nilsystems, as compared with arbitrary dynamical systems, is that ergodicity can automatically be upgraded to unique ergodicity:
Theorem 2. Let (X,T) be an ergodic nilsystem. Then (X,T) is also uniquely ergodic. Equivalently, for every , the orbit is equidistributed.
Exercise 8. By inducting on step and adapting the proof of Proposition 3 from Lecture 9, prove Theorem 2.
– a Ratner-type theorem –
A subnilsystem of a nilsystem is a compact subsystem which is of the form for some and some closed subgroup . One easily verifies that a subnilsystem is indeed a nilsystem.
From the above theorems we quickly obtain
Corollary 1 (Dichotomy between structure and randomness) Let (X,T) be a nilsystem with group G connected and simply connected, and let . Then exactly one of the following statements is true:
- The orbit is equidistributed.
- The orbit is contained in a proper subnilsystem (Y,S) with group H connected and simply connected, and with dimension strictly smaller than that of G.
Proof. It is clear that 1. and 2. cannot both be true. Now suppose that 1. is false. By Theorem 2, this means that (X,T) is not ergodic; by Theorem 1, this implies that the Kronecker system is not ergodic. Expanding functions on into characters and using Fourier analysis, we conclude that there is a non-trivial character which is -invariant. If we let be the canonical projection, then is a continuous homomorphism, and the kernel H is a closed connected subgroup of G of strictly lower dimension. Furthermore, Hx is equal to a level set of and is thus compact. Since is invariant, we see that for all n, and the claim follows.
Iterating this, we obtain
Corollary 2 (Ratner-type theorem for nilmanifolds) Let (X,T) be a nilsystem with group G connected and simply connected, and let . Then the orbit is equidistributed in some subnilmanifold (Y,S) of (X,T). (In particular, this orbit is dense in Y.) Furthermore, Y= Hx for some closed connected subgroup H of G.
Remark 7. Analogous claims also hold when G is not assumed to be connected or simply connected, and if the orbit is replaced with a polynomial orbit ; see this paper of Leibman for details (and this followup paper for the case of -actions. In a different direction, such discrete Ratner-type theorems have been extended to other unipotent actions on finite volume homogeneous spaces by Shah. Quantitative versions of this theorem have also been obtained by Ben Green and myself.
[Update, June 29: minor corrections.]