What are nilspaces?

Yonatan Gutman

Table of Contents


A cubespace is a structure consisting of a compact metric space X, together with closed sets Ck(X)X2k for each integer k0, satisfying certain axioms that we will recall later. We think of Ck(X) as determining when a collection of 2k points of X form a “k-cube”. The structure (X,Ck(X)) is further called a nilspace if it also satisfies certain extra rigidity conditions. The notion of nilspaces has its origins in the work of Host and Kra [HK08], where these objects appeared under the name of “parallelepiped structures”. The study of these objects was furthered by Antolín Camarena and Szegedy [ACS12] (for an exposition see [Can16a, Can16b]), who in the same work formulated a strong structure theorem for nilspaces. The gist of the structure theorem is that nilspaces arise as inverse limits of nilmanifolds X=G/Γ (here G is a nilpotent Lie group and ΓG is a discrete cocompact subgroup); i.e. they come from compact homogeneous spaces of nilpotent Lie groups.

The theory of nilspaces is a useful tool in the area of higher order Fourier analysis (see [Tao12, Gow17]), and in particular, forms the basis of Szegedy’s approach [Sze12] to proving the inverse theorem for the Gowers norms (another approach being due to Green, Tao and Ziegler [GTZ12]).

Nilspaces can also be used in topological dynamics and ergodic theory. For example together with Freddie Manners and Péter Varjú we used them to generalize a result of Host, Kra and Maass [HKM10] characterizing the largest pronilfactor of a minimal group action ([GMV16a, GMV16b, GMV16c]).

Obstructions to Gowers uniformity

A fundamental concept in additive combinatorics and in higher order Fourier analysis is that of Gowers norms (introduced by Gowers in [Gow01]). Given a function f:/N for N a prime (say), the Gowers norm of order 3 fU3 is defined in terms of an average over cube or parallelepiped configurations in /N, e.g.


where C3(/N)(/N)8 consists of all tuples


for x,h1,h2,h3/N.

Following the Introduction of [GMV16a] quite closely we will (very) informally discuss why nilspaces of degree 2 are obstructions for U3 to be “small”. For notational simplicity we focus only on Gowers norms of order 3, but the argument works for all orders. Let us start by sketching the proof of why a function which correlates with a nilsequence (see below) has “large” Gowers norm (following [GT08, Proposition 12.6]):

  1. Given a nilmanifold G/Γ (with suitable additional structure) there is also a notion of cubes on G/Γ, given by a construction due to Host and Kra ([HK05, HK08]) - see details below. Specifically, suppose G is a 2-step nilpotent Lie group and Γ a discrete co-compact subgroup; then there is a closed subset C3(G/Γ)(G/Γ)8 somewhat analogous to the parallelepipeds in an abelian group.

  2. There is a plentiful supply of maps p:/NG/Γ which send cubes to cubes; that is, p(c)C3(G/Γ) for any cC3(/N) (with p applied pointwise).

  3. The cubes on G/Γ satisfy a corner constraint: given cC3(G/Γ), if we know c000,,c110 then the last vertex c111 is uniquely determined by the others.

By a nilsequence on /N we mean a function of the form ϕ=Fp where p is as above and F:G/Γ is Lipschitz. We thus prove: Suppose ϕ=Fp is a nilsequence with F1 and F¬0 and let δ>0. There exist ϵ=ϵ(F,δ) so that if f:/N is such that |f|1 and |f,ϕ||1Nx=0N-1f(x)ϕ(x)|δ, then fU3ϵ. The key point is that the lower bound on fU3 depends only on the choice of F (and so implicitly of G/Γ) and on δ; not on N or p.

[Proof sketch] By property (3), there is a closed subset Y(G/Γ)7 and a function τ:Y(G/Γ) such that


Hence we get a continuous function Fτ on Y. By the Tietze extension theorem, we can extend this to a bounded continuous function H on (G/Γ)7. Any continuous function on a product space can be approximated (up to a small error in L) by a finite sum of products of functions on the factors: that is, we can decompose


for some bounded continuous functions Rj(i):G/Γ, and some bounded continuous Herr:(G/Γ)7 such that Herr=oF;k(1).

Now, for any x,h1,h2,h3 in /N we have that (p(x),p(x+h1),p(x+h2),p(x+h1+h2),) is in C3(G/Γ), and so






Since |f,ϕ| is bounded away from zero, we deduce that


is bounded away from zero for some i{1,,k} (after choosing k appropriately in terms on F and δ). But this expression is a “Gowers inner product” of eight functions, and by the Gowers–Cauchy–Schwarz inequality (essentially multiple applications of Cauchy–Schwarz, see [Tao12, Exercise 1.3.19]), this quantity is bounded above by


and noting that Rj(i)pU3Rj(i) which is bounded, we get a lower bound on fU3 as required.

The key point is that the only properties of nilmanifolds, nilsequences etc. that we have used are those described in (1),(2) and (3) above. So we have in fact shown:

If X is any compact metric space equipped with some suitable notion of “cubes” as in (1), having cube-preserving maps p:/NX as in (2), and satisfying a corner constraint as in (3), then functions of the form Fp, where F:X is continuous, obstruct Gowers uniformity on /N in the sense of the above claim.

Let us refer to such a space informally for now as a “nil-object” (the formal notion of a nilspace will be introduced later). Then the above observation can be summarized as follows: Any “nil-object” is an obstruction to Gowers uniformity.

The inverse theorem for the Gowers norms states that given δ>0 there exist ϵ=ϵ(δ) such that if f:/N, |f|1 has fU3δ then |f,ϕ|ϵ for some nilsequence ϕ whose “complexity” is bounded in terms of δ. This tends to suggest that any “nil-object” X must be very closely related to an actual nilmanifold G/Γ.

The conclusion of work of Szegedy [Sze12] is that it is possible to go in the other direction. He argues that all functions f with fU3 somewhat large correlate with something of the form Fp where p:/NX and F:X is continuous, for some space X equipped with a notion of cubes, and some cube-preserving p, obeying some fairly reasonable additional axioms. Assuming this, we conclude: The class of all “nil-objects” corresponds precisely to the obstructions to Gowers uniformity.

Hence, the inverse theorem for the Gowers norms is essentially equivalent to classifying nil-objects X, showing in effect that they are all – essentially – nilmanifolds. This structural result is the goal of [ACS12] (this is also proven by somewhat different methods in [GMV16a, GMV16b, GMV16c].).

The nilspace axioms

At this stage you are probably curious what are these “nil-objects” from the previous section. We list the axioms without further comments and in the next section give some examples. A map ϕ=(ϕ1,...,ϕk):{0,1}l{0,1}k is called a morphism of discrete cubes if each coordinate function ϕj(ω1,...,ωl) equals to either 0, 1, ωi or 1-ωi for some 1il. Let X be a metric space and for each integer l0 let Cl(X)X{0,1}l be a closed set. We say that (X,C(X)) is a cubespace if C0=X and cϕCl(X) for any morphism of discrete cubes ϕ:{0,1}l{0,1}k and any cCk(X). We refer to this property as cube invariance. We call the elements of Cl(X) cubes of order l (in short l-cubes). We call the points c(ω),ω{0,1}l the vertices of c.

Let X be a cubespace and let f:X{0,1}l{1} be a map. We call f an l-corner if f|{ωi=0} is an (l-1)-cube for all 1il. We say that the cubspace X has s-completion if any s-corner f can be extended to a s-cube, that is to say, if there is a cube cCs(X) such that c|{0,1}s{1}=f. We say that X is fibrant if it has s-completion for all s. We say that a cubespace X has (s+1)-uniqueness, if c1|{0,1}s+1{1}=c2|{0,1}s+1{1} imples c1=c2 for any two (s+1)-cubes c1,c2Cs+1(X).

A cubespace X is a nilspace of degree s if it is fibrant and has (s+1)-uniqueness. We say that a cubespace X is a nilspace if it is a nilspace of degree s for some s.

Examples of nilspaces

A simple example is given by the following construction. Let A be a compact abelian group. We write 𝒟s(A) for the cubespace defined by requiring that cC(𝒟s(A)) if and only if


holds for any morphism of discrete cubes ϕ:{0,1}s+1{0,1}, where we write |ω|=1is+1ωi for ω{0,1}s+1. One can prove that 𝒟s(A) is a nilspace of degree s (see [GMV16a, Example A.9]). A more advanced example is given by the following construction of the

Host–Kra nilspaces. Let G be a (metric) topological group. We call a chain of closed subgroups


a filtration of degree s if [Gi,Gj]Gi+j for all i,j0, adopting the convention that Gi={1} for all is+1 (e.g. the lower central series of a nilpotent group G). We write G as a shorthand to denote a group G equipped with a filtration {Gi} and call it a filtered group.

We define the HK(G) for each to be the subgroup of G{0,1} generated by the elements of the form [g]F, where F{0,1} is a face of codimension i for some 1i, gGi and [g]F is the element of G{0,1} given by [g]F(ω)=g if ωF and [g]F(ω)=id otherwise. It is not hard to verify that (G,HK(G)) is a cubespace.

Let X=G/Γ be a nilmanifold (that is G is a nilpotent Lie group and ΓG is a discrete cocompact subgroup). Let G be a filtration of degree s on G. Define the cubespace HK(G)/Γ(X,C(G/Γ)) by


One can prove that HK(G)/Γ is a nilspace of degree s (see [GMV16a, Proposition 2.6]).


[ACS12] Omar Antolín Camarena and Balázs Szegedy. Nilspaces, nilmanifolds and their morphisms. Preprint. http://arxiv.org/abs/1009.3825, 2012.

[Can16a] Pablo Candela. Notes on compact nilspaces. arXiv preprint arXiv:1605.08940, 2016.

[Can16b] Pablo Candela. Notes on nilspaces: algebraic aspects. arXiv preprint arXiv:1601.03693, 2016.

[GMV16a] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces I. Preprint. arxiv.org/abs/1605.08945, 2016.

[GMV16b] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces II: Representation as nilmanifolds. Preprint. arxiv.org/abs/1605.08948, 2016.

[GMV16c] Yonatan Gutman, Freddie Manners, and Péter P. Varjú. The structure theory of nilspaces III: Inverse limit representations and topological dynamics. Preprint. arxiv.org/abs/1605.08950, 2016.

[Gow01] William T Gowers. A new proof of Szemerédi’s theorem. Geometric and functional analysis, 11(3):465–588, 2001.

[Gow17] W Gowers. Generalizations of Fourier analysis, and how to apply them. Bulletin of the American Mathematical Society, 54(1):1–44, 2017.

[GT08] Ben Green and Terence Tao. An inverse theorem for the Gowers U 3 (G) norm. Proceedings of the Edinburgh Mathematical Society (Series 2), 51(01):73–153, 2008.

[GTZ12] Ben Green, Terence Tao, and Tamar Ziegler. An inverse theorem for the Gowers U s+1 [N]-norm. Ann. of Math. (2), 176(2):1231–1372, 2012.

[HK05] Bernard Host and Bryna Kra. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2), 161(1):397–488, 2005.

[HK08] Bernard Host and Bryna Kra. Parallelepipeds, nilpotent groups and Gowers norms. Bull. Soc. Math. France, 136(3):405–437, 2008.

[HKM10] Bernard Host, Bryna Kra, and Alejandro Maass. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math., 224(1):103–129, 2010.

[Sze12] Balázs Szegedy. On higher order Fourier analysis. Preprint. http://arxiv.org/abs/1203.2260, 2012.

[Tao12] Terence Tao. Higher order Fourier analysis, volume 142. American Mathematical Soc., 2012.