In the last few years, there has been a surge of activity related to high-dimensional expanders (HDXs). Loosely speaking, high-dimensional expanders are a high-dimensional generalization of classical graph expanders. Depending on which definition of graph expansion is generalized, there are several different (and unfortunately, many a time mutually inequivalent) definitions of HDXs. For the purpose of this note, we will restrict ourselves to the spectral definition of HDXs (see Definition 2.5). Lubotzky, Samuels and Vishne [LSV05a, LSV05b] constructed high-dimensional analgoues of the Ramanujan expanders of Lubotzky, Philips and Sarnak [LPS88], which they termed Ramanujan complexes. These Ramanujan complexes have several desirable properties and gave rise to the first construction of constant degree spectral HDXs. The Ramanujan graphs have the nice property that they are simple to describe, however their proof of expansion is extremely involved. The Ramanujan complexes, on the other hand, are both non-trivial to describe as well as to prove their high-dimensional expansion property. Subsequently Kaufman and Oppenheim [KO18] gave an extremely elegant and elementary construction of spectral HDXs using elementary matrices. Despite their construction being elementary and simple, the proof of expansion, though straightforward, requires some knowledge of some representation theory of the specific groups involved in the construction. The purpose of this exposition is to give an alternate elementary proof of the expansion of the Kaufman-Oppenheim HDX construction.
The underlying graph of a HDX (even a onesided spectral HDX) is a two-sided spectral expander. Thus, this construction has the added advantage that it yields an elementary construction (accompanied with a simple proof) of a standard two-sided spectral expander (though not an optimal one).
We begin by recalling what a simplicial complex is.
Definition 2.1 (Simplicial complex).
A simplicial complex over a finite set is a collection of subsets of with the property that if then any is also in .
For all , define Thus, if is non-empty, then .
The elements of are called simplices or faces. The elements of , and are usually referred to as vertices, edges and triangles respectively.
The graph defined by and is called the -skeleton of the complex. More generally, for any , the -skeleton of the complex is the sub-complex .
The dimension of the simplicial complex defined as the largest such that (which consists of faces of size ) is non-empty.
The simplicial complex is said to be pure if every face is contained in some face in , where .
For a face , the link of , denoted by , is the simplicial complex defined as
Thus, a graph is just a simplicial complex of dimension one with and . We will deal with weighted pure simplicial complexes where the weight function satisfies a certain balance condition.
Definition 2.2 (weighted pure simplicial complexes).
Given a -dimensional pure simplicial complex and an associated weight function , we say the weight function is balanced if the following two conditions are satisfied.
A weighted simplicial complex is a pure simplical
complex accompanied with a balanced weight function . If no
weight function is specified, then we work with the balanced weight
function induced by the uniform distribution on the set
induced by the uniform distribution on the setof top faces.
For a face , the balanced weight function associated with the link is the restricted weight function, suitable normalized, more precisely .
) states that the weight function can be interpreted as a family of joint distributionswhere
is a probability distribution on. The distribution is specified by the first condition in (2.3) while the second condition implies that the weight distribution is the distribution on obtained by picking a random according to and then removing elements uniformly at random.
We now recall the classical definition of what it means for a graph to be a spectral expander.
Definition 2.4 (spectral expander).
Given an undirected weighted graph on vertices, let be its normalized adjacency matrix given as follows:
Let be the eigenvalues of with multiplicities in non-increasing order111By the balance condition, satisfies . The matrix is self-adjoint with respect to the inner product since . Hence, has real eigenvalues which can be obtained using the A.2.. We denote the second largest eigenvalue of as .
is said to be a -spectral expander if .
is said to be a -onesided-spectral expander if .
This spectral definition of expanders is generalized to higher dimensional simplicial complexes as follows.
Definition 2.5 (-spectral HDX).
A weighted simplicial complex of dimension is said to be a -spectral HDX222These are sometimes also referred to as -link HDXs or -local-expanders. if for every and , the weighted -skeleton of the link is a -spectral expander.
A weighted simplicial complex of dimension is said to be a -onesided spectral HDX if for every and , the weighted -skeleton of the link is a -onesided-spectral expander.
Descent Theorem 2.5 ([Opp18]).
Suppose is a -dimensional weighted simplicial complex with the following properties.
For all , the link is a -(onesided)-spectral expander.
The -skeleton of every link is connected.
Then, is a -(onesided)-spectral HDX.
3 Coset complexes
In this section, we construct certain special simplicial complexes based on a group and its subgroups, that are called coset complexes. For a basic primer on group theory, see Appendix B
Definition 3.1 (coset complex).
Let be a group and let be subgroups of . The coset complex is a -dimensional simplicial complex defined as follows:
The vertices, , consists of cosets of and we shall say cosets of are of type .
The maximal faces, , consists of -sets of cosets of different types with a non-empty intersection. That is,
An equivalent way of stating this is that if and only if there is some such that for all .
The lower dimensional faces are obtained by down-closing the maximal faces. Hence, for , if and only if for all and
We shall call the set the type of this face.
The dimension of this complex is .
The weight function we will use is the one induced by the uniform distribution on the top face .
A simplicial complex constructed this way is partite in the sense that each maximal face consists of vertices of distinct types.
if and only if .
() Say for and . Then .
() If for and , then . ∎
Lemma 3.3 (Criterion for connected -skeletons).
The -skeleton (underlying graph) of is connected if and only if .
() Since there is always an edge between and for , it suffices to show that is connected to for an arbitrary . Suppose, for an arbitrary element , we have where and for each . We might wlog. assume that (a) (otherwise set ) and (b) if , then (since otherwise we might then have worked with as ).
Then, we get the following path connecting and
Note that, due to Observation 3.2, each
successive pair of cosets are connected by an edge in the simplicial
complex. Now, since is adjacent to (as ), we have that is connected to .
() For an arbitrary , since the -skeleton is connected we have a path
By Observation 3.2, for every , we have . Therefore,
Structure of links of the coset complex:
For any set , define the group ; let . The following lemma shows that the links of a coset complex are themselves coset complexes.
For any of type , the link is isomorphic to the simplicial complex defined by .
It suffices to prove this lemma for as links of higher levels can be obtained by inductive applications of this case.
Observe that if is any element of , then if and only if the tuple . Therefore, the link of a coset is isomorphic to the link of the coset . Thus, it suffices to prove the lemma for links of the type for some .
Let be the coset , without loss of generality. The vertices of the link, , are cosets of that have a non-empty intersection with . Note that any non-empty intersection of a cosets with is itself a coset of the intersection subgroup in . Therefore, the vertices of the link are in bijective correspondence with cosets of .
The maximal faces in that contain the coset are precisely -sets of cosets with a non-empty intersection and hence
which are precisely the maximal faces of the coset complex . This establishes the isomorphism between and . ∎
4 A concrete instantiation
The simplicial complex of Kaufman and Oppenheim [KO18] is a specific instantiation of the above coset complex construction. We will need some notation to describe their group.
Let denote the ring . This is a ring whose elements can be identified with polynomials in of degree less than (where addition and multiplication are performed modulo ). We will think of as some fixed prime power, a formal variable and as a growing integer.
For with and an element , we define the to be the elementary matrix with ’s on the diagonal and on the -th entry.
For the sake of notational convenience, we shall abuse this notation and use etc. to refer to by wrapping around if necessary. For example, refers to .
We are now ready to describe the groups in the construction.
Each is generated by elementary matrices that have ’s on the diagonal and an arbitrary linear polynomial in one entry of the generalised diagonal .
It so happens that the group generated by the subgroups is , the group of matrices with entries in whose determinant is (in ). This is a non-trivial fact. All we will need is the simpler fact that grows exponentially with (for fixed and ) while the size of the groups are functions of and (and independent of ). This will follow from the sequence of observations and lemmas developed in the following section.
4.1 Explicit description of the groups
The following is an easy consequence of the definition of .
Sum: If , then .
Product: If and , then the commutator333The commutator of two elements , denoted by is defined as . (B.2) behaves as follows.
Therefore, we have that
and hence can be generated for any . This in particular implies that is at least . On the other hand, the size of depends only on and is independent of . The lemma below describes ; the other ’s are just rearrangements of rows and columns in .
Lemma 4.2 (Explicit description of ).
The group consists of matrices of the following form:
Therefore, we can obtain a crude bound of . In fact, we can generalise the above definition to define the group for any as follows:
These groups can also be explicitly described.
Lemma 4.3 (Explicit description of ).
For any , group is the set of all matrices of the form
Any can be expressed as where each , for some linear polynomial , with . Then,
From the structure of each , any nonzero contribution from the RHS must involve either , or if . This forces that the only entries of that are nonzero, besides the diagonal, are at with none of in .
In the case when , the above argument also shows that the entry has degree at most . Furthermore, Observation 4.1 shows that for an arbitrary polynomial of degree at most . From this, we can deduce that the structure of is exactly as claimed. ∎
From the explicit structure of the groups, we have the following corollary.
Corollary 4.4 (Intersections of ’s).
For any ,
In other words, the group generated by the intersection of generators equals the group intersection.
The above corollary tells us that we can drop the tilde notation and use for .
4.2 Connectivity of this complex
Let with . Then,
It is clear that is a superset of the RHS. It only remains to show that the other containment also holds. To see this, consider an arbitrary generator of . Since and , there is some . Therefore, and hence is generated by the RHS. ∎
For the coset complex defined by the above groups, the -skeleton of every link is connected.
5 Spectral expansion of the complex
In this section we prove that the coset complex is a good spectral HDX. The Section 2 states that it suffices to show that the -dimensional links of faces in are good spectral expanders.
5.1 Structure of -dimensional links
One dimensional links of the coset complex constructed are links of of size exactly (which are elements of . Any such can be written as for with and . Since the link of is isomorphic to the link of , we might as well assume that . These happen to be of two types depending on whether and are consecutive or not.
Consider where and are not consecutive (i.e. ). Then the -dimensional link of is a complete bipartite graph.
Note that since , we have by 4.1. Hence, these two elements commute.
The link of corresponds to the coset complex where
Thus, the groups and commute with each other and hence any element of can be written as where and . 3.2 implies that the resulting graph is the complete bipartite graph. ∎
The interesting case is when . Without loss of generality, we may focus on the link of . This corresponds to the coset complex where
Hence, it suffices to focus on the first three rows and columns of these matrices as the rest of them are constant. Written down explicitly,
Therefore, each coset of and in has a unique representative of the form
respectively, where is a linear polynomial and is a quadratic polynomial in .
For linear polynomials and quadratic polynomials , we have that
By Observation 3.2, the cosets have a non-empty intersection if and only if
which happens if and only if which is the same as . ∎
Therefore, the -dimensional link is the bipartite graph with left and right vertices identified by pairs where and are linear and quadratic polynomials in respectively, with (by associating with the tuple on the left, and with the tuple on the right).
Note that is an undirected, -regular bipartite graph with vertices on each side. It suffices to show that is a good expander.
5.2 A related graph
The following graph is the “lines-points” or the “affine plane” graph used by Reingold, Vadhan and Wigderson [RVW05] (as the base graph in construction of constant-degree expanders, using the zig-zag product). Let be a finite field. Consider the bipartite graph defined as follows:
The -regular bipartite graph is a -onesided-spectral expander.
Consider the graph restricted to the vertices in . It is easy to see that
Therefore, the adjacency matrix of (restricted to ) can be written (under a suitable order of listing vertices)
Hence the unnormalized second largest eigenvalue of is and hence we have that the normalized second largest eigenvalue of is . ∎
5.3 Relating the graph with
Set so that for some irreducible polynomial of degree exactly . Therefore, each element in is expressible as for some . Thus, the graph defined above, for this setting of , is a -regular bipartite graph with vertices on either side.
Let , which is a subset of and , respectively, of size each.
The induced subgraph of on is exactly the graph described earlier.
Note that if and only if
However, since the above equation has degree at most , we have
and the first equation is exactly the adjacency condition of the graph . Hence, the induced subgraph of on is indeed the graph . ∎
Normally, induced subgraphs of expanders need not even be connected. However, the following lemma shows that there are some instances when we may be able to give non-trivial bounds on .
Suppose is a -regular, undirected graph that is an induced subgraph of a -regular graph . Then,
The graph corresponding to the -dimensional links of is an onesided -spectral expander.
The graph is a bipartite, -regular graph with and is a -regular graph that is an induced subgraph of . Hence, by Lemma 5.5,
The final expansion bounds
From the corollary above, we obtain the following theorem of Kaufman and Oppenheim.
Theorem 5.7 ([Ko18]).