Ultraproducts of quasirandom groups with small cosocles
-
Yilong Yang
Abstract
A D-quasirandom group is a group without any non-trivial unitary representation of dimension less than D. Given a sequence of groups with increasing quasirandomness, it is natural to ask if the ultraproduct will end up with no finite dimensional unitary representation at all. This is not true in general, but we answer this question in the affirmative when the groups in question have uniform small cosocles, i.e., their quotients by small kernels are direct products of finite simple groups.
Two applications of our results are given, one in triangle patterns inside quasirandom groups and one in self-bohrifying groups. Our main tools are some variations of the covering number for groups, different kinds of length functions on groups, and the classification of finite simple groups.
1 Introduction
As an indirect consequence of Kassabov, Lubotzky and Nikolov’s paper [12], the following theorem about non-abelian finite simple groups is true.
An ultraproduct of non-abelian finite simple groups is either finite simple, or has no finite-dimensional unitary representation other than the trivial one.
Definitions related to ultraproducts are presented in Section 2 for those unfamiliar with them.
In this paper, we shall show that non-abelian finite simple groups are not the only kind of groups exhibiting such behavior. It turns out that such behavior has a very close link to the notion of quasirandom groups, defined by Gowers [7], and the notion of minimally almost periodic groups, defined by von Neumann and Wigner [13]. All representations considered in this paper are over
For a positive integer D, a group G is D-quasirandom if it has no non-trivial unitary representation of dimension less than D.
An infinite group is minimally almost periodic if it has no non-trivial finite dimensional unitary representation.
A group is minimally almost periodic iff it is D-quasirandom for all D. Then it is natural to wonder whether some sort of limit of increasingly quasirandom groups would give us a minimally almost periodic group. One such limit to consider is the ultraproduct.
We will prove the existence of classes of groups with similar results to Theorem 1. The main theorem is the following Theorem 5.
For a group G, we define its cosocle
Let n be any positive integer. Let
For any sequence of groups in
Quasirandom groups were first introduced by Gowers to find groups with no large product-free subset. They can be seen as stronger versions of perfect groups.
Example 6 (Gowers [7])
A group (not necessarily finite) is 2-quasirandom iff it is perfect. The reason is that a non-perfect group has a non-trivial abelian quotient, which in turn has a non-trivial homomorphism into
A finite perfect group with no normal subgroup of index less than n is at least
In particular, a non-abelian finite simple group G is at least
Conversely, any D-quasirandom group must have more than
The alternating group
Morally, ultraproducts preserve all local properties at the scale of elements. In particular, all element-wise identities are preserved. But global properties of a group, like being finite or finitely generated, might be lost after taking ultraproducts. So one may wonder if a non-principal ultraproduct of increasingly quasirandom groups is always minimally almost periodic. In other words, we want to investigate if quasirandomness can be captured by element-wise properties. This turns out to be false. In particular, we have the following counterexample, pointed out by László Pyber.
We recall that a group G (not necessarily finite) is 2-quasirandom iff G is perfect. We claim that there is a sequence of
Using the construction of Holt and Plesken [11, Lemma 2.1.10], one may construct a finite perfect group
Let
However, a recent paper by Bergelson and Tao [3] proved the following theorem, which sheds some new light on this inquiry:
Theorem 8 ([3, Theorem 49 (i)])
The ultraproduct
Inspired by this, we make the following definitions:
A class
A class
A Q.U.P class is automatically a q.u.p. class. It is like an effective version of q.u.p. class, where we are able to keep track of the amount of quasirandomness passed down to the ultraproduct.
In this paper, the proof of Theorem 5 in fact shows that the class
The following classes are Q.U.P.
The class
The class
The class
All Q.U.P. classes must have a uniformly bounded commutator width, i.e., every element can be written as a product of uniformly bounded number of commutators. In view of this, the following conjecture was suggested by László Pyber.
For any integer n, the class of perfect groups with commutator width
So far, we do not know if there is a non-Q.U.P but q.u.p. class of groups.
Some applications of our results have already been found. In a paper in preparation by Bergelson, Robertson and Zorin-Kranich [2, Theorem 1.12], it is shown that a sufficiently quasirandom group in a q.u.p. class will have many “triangles”. As another application, one may also use our method to find many examples of self-bohrifying groups. Both applications will be explained in Section 8 of this paper.
Here we shall briefly outline the sections of this article:
A model case of the alternating groups to illustrate the general idea (Section 3).
A group with a nice covering property is very quasirandom (Section 4).
Covering properties can ignore small cosocles (Section 5).
Quasirandom finite quasisimple groups have nice covering properties (Section 6).
Applications of our results (Section 8).
2 Definitions relating to ultraproducts
A filter on
if
if
An ultrafilter is a filter that is maximal with respect to the containment order. A non-principal ultrafilter is an ultrafilter that contains no finite subset of
Given a sequence of groups
which is clearly a normal subgroup of G. Then we call
An ultrafilter ω is principal (i.e., not non-principal) iff we can find an element
The particular choice of ultrafilter is not that important. As long as we fix a non-principal ultrafilter, then all the discussion in the rest of the paper will be true for the ultraproduct of this ultrafilter.
Ultraproducts have an interesting property, given by Łoś’ Theorem. Given an ultraproduct
In particular, this implies that behaviors at the scale of elements are preserved. We shall not need Łoś’ Theorem in this paper, but it could be used as an alternative to Proposition 64.
3 The class of alternating groups
Let
3.1 Quasirandom alternating groups have nice covering properties
For any subsets
And we define
An element g of a group G is said to have covering numberK if its conjugacy class
Let m be any positive integer or
A group G has the covering property
Note that we use
An even permutation
Lemma 18 (Brenner [4, Lemma 3.05])
If an even permutation
For any
Proof.
Pick any odd prime
Since
Since
Now clearly
For any
3.2 Covering properties passes to ultraproducts and implies quasirandomness
Let
Proof.
Since non-principal ultraproducts ignore finitely many exceptions in the sequence
For each
Pick any
We now state a special case of Proposition 27, proven in Section 4.
There is a function
The class of alternating groups is a Q.U.P. class.
Proof.
For any
4 Covering properties imply quasirandomness
This section is devoted to obtaining some element-scale properties that guarantee the quasirandomness of a group.
An element g of a group G is said to have symmetric covering numberK if
Let m be a positive integer or
A group G has the symmetric covering property
A group G has the (symmetric) covering property
A pair of elements
Let
A group G has the symmetric double covering property
A group G has the symmetric double covering property
Suppose
Any symmetric covering property is always weaker than the corresponding non-symmetric covering property.
Any group with the symmetric covering property
In our definition of the symmetric double covering properties, since
By imitating the definition of the symmetric double covering properties, one can in fact define the symmetric n-tuple covering properties for groups. As n grows larger and larger, the corresponding covering properties will become weaker and weaker. Note that most results throughout this paper would still hold by replacing the symmetric double covering properties by the symmetric n-tuple covering properties, though for our purpose here, the symmetric double covering properties are enough.
The proof of Proposition 27 will be the main part of this section. Let us first state the proposition and some corollaries.
Proposition 27 (Local criterion for quasirandomness)
There is a function
such that, for any
We shall fix this function f from now on.
For any
For any
We note here that a partial converse, Corollary 66, of the above result is true. That is, quasirandomness implies a nice covering property mod cosocle. The proof of this converse will be presented in Section 7.
We shall first explore some geometric structures of
The Hilbert–Schmidt norm of an n-by-n complex matrix A is
The following statements hold.
The Lie group
This metric is bi-invariant in the sense that
for all
This metric induces a Haar measure, and the volume of
We shall denote this constant by
Under the metric d,
There is a function
Let G be any group. A non-negative function
A pseudo-length function is a non-negative function
Let G be a group, and suppose
Proof.
For any
The function
Proof.
Let
Positivity:
Clearly
Symmetry: We have
Conjugate invariance: We have
Triangle inequality: We have
We shall use
For any
Proof.
This follows from a volume packing argument.
Since our metric is bi-invariant, each ball of radius
Now for any m points in
Any non-trivial cyclic subgroup of
Proof.
Let A be any non-trivial element of
Then we know
Now suppose A has infinite order. Let
Proof of Proposition 27.
For any
For any unitary representation
Now consider the points
So we have
To sum up, there are elements
Now pick
(Say
So, a group with the symmetric double covering property
Note that the above argument proves Proposition 27 for all groups, not necessarily finite. However, if one only needs to prove Proposition 27 for finite groups, and only for the covering property
5 Covering properties and the cosocle
In this section, we will show that a certain nice covering property mod cosocle is equivalent to a weaker covering property of the whole group.
Let G be a group, and let N be a normal subgroup of G contained in its cosocle. Let C be a conjugate invariant symmetric subset of G such that
Proof.
Suppose
But since N is in the cosocle, it is contained in M. So
This is a contradiction. ∎
Let G be a group with the symmetric double covering property
Proof.
Find
Then by assumption, C is mapped surjectively onto
Now N contains exactly n conjugacy classes of G. I claim that
We proceed by induction. As a convention we define
Now assume the statement is true for some
Suppose for contradiction that
So
So,
Finally, since
Let G be a group with the symmetric covering property
Proof.
Same strategy as Proposition 40. ∎
6 Quasirandom finite simple groups have nice covering properties
In this section we shall show that, for finite quasisimple groups, large quasirandomness will imply a nice covering property. We shall first deal with finite simple groups of bounded ranks in Section 6.1. Then we shall deal with the case of alternating groups in Section 6.2. Finally, we shall deal with finite simple groups of large ranks by embedding alternating groups into them in Section 6.3. The classification of finite simple groups is used in this section.
For a finite quasisimple group G, we define its rank
When the only simple quotient of G is abelian or sporadic, then
When the only simple quotient of G is
When the only simple quotient of G is a group of Lie type, then
6.1 Finite simple groups of bounded ranks
Lemma 43 (Stolz and Thom [15, Proposition 3.8])
There is a function
such that, in any finite simple group of Lie type of rank
We shall fix this function
Lemma 44 (Babai, Goodman and Pyber [1, Proposition 5.4])
Let k be any positive integer. Then for any finite simple group G, if
Let G be a finite simple group of rank
Proof.
By choosing D to be larger than some absolute constant, a D-quasirandom group G cannot be an abelian group, a sporadic group, or an alternating group of rank
Recall that any D-quasirandom group must have more than
Let G be a finite quasisimple group of rank
Proof.
If a quasisimple group is D-quasirandom, then the simple group it covers is D-quasirandom. Therefore, it is enough to show that, if a finite simple group G has the covering property
Let Z be the center of
6.2 Alternating groups
Let G be a quasisimple group over an alternating group. Then for any
Proof.
If G is D-quasirandom for some large D, then the alternating group it covers must be
6.3 Finite simple groups of large ranks
The goal of this subsection is to prove the following proposition.
There is an absolute constant
By the classification of finite simple groups, a finite simple group of rank larger than some absolute constant will have to be a classical finite simple group of Lie type or an alternating group. Any classical finite simple group of Lie type is in one of the following four classes:
The projective special linear groups
The projective symplectic groups
The projective special unitary groups
The projective Omega groups
The plus or minus signs indicate different quadratic forms used to obtain the groups in even dimensions. For large enough n,
The above statements can be found in any standard textbook in classical groups (e.g., see [8]). It is enough to show Proposition 48 for
We start by analyzing a length function for groups of Lie type over finite fields.
Let g be an
Then the Jordan length of g is
Let G be any subgroup of
Proof.
Non-negativity: For any
So
Symmetry: For any
As a result,
So
Conjugate-invariance: For any
As a result,
So
Triangle inequality: For any
So we know
Since this is true for all
So
Given an
Proof.
For any
So
Lemma 52 (Stolz and Thom [15, Lemma 3.11])
There is an absolute constant
In short, elements of large Jordan length will automatically have small covering number.
The next step is to identify subgroups of these quasisimple groups of Lie type isomorphic to the alternating groups. A key step is to treat elements in alternating groups as matrices, namely permutation matrices. These are the matrices with exactly one entry of value 1 in each column and in each row, and 0 for all other entries. Such an
If P is an
Proof.
By cycle decomposition, after a change of basis in the vector space, P will be a direct sum of many cyclic permutation matrices. By Lemma 51, it is enough to prove the case when P is a single cycle of length n, and show that
Since P is a single cycle of length n, its eigenvalues in the algebraic closure of F are precisely all the n-th roots of unity, with multiplicity 1 for each root of unity. So
There is an absolute constant
Proof.
Let
Pick any odd prime
For any finite quasisimple group of Lie type of
The last inequality follows because
So this element will have covering number
For any
There is an absolute constant
For any finite quasisimple group of Lie type of
Let
Let
Proof.
The strategy is identical to Proposition 54. Just take
A vector space V is a non-degenerate formed space if it has a non-degenerate quadratic form Q (the orthogonal case), or a non-degenerate alternating bilinear form B (the symplectic case), or a non-degenerate Hermitian form B (the unitary case).
Lemma 58 (Witt’s Decomposition Theorem)
Let V be any non-degenerate formed space over a finite field F. Then we have an orthogonal decomposition
Proof.
These are standard facts in the geometry of classical groups (see [8]). ∎
For a non-degenerate formed space, the special isometry group, i.e., the group of isometries of determinant 1, contains an alternating group in one of the ways described by Proposition 56.
Proof.
Let V be any finite-dimensional non-degenerate formed space over any finite field F. Then we have an orthogonal decomposition
Then let
Now clearly
Finally, since P is a permutation matrix for an even permutation, it has determinant 1. Therefore the above embedding of
For any
For any
Proof.
Embed
Proposition 48 is proven by putting Corollary 55, Corollary 60 and Corollary 61 together.
7 Proof of Theorem 5
The results of Section 6 can be summarized into the following useful lemma.
For any integer D and any constant c, we can find integers
Proof.
Let
Set
In all cases, a
In the above proof, one cannot substitute the double covering properties with the covering properties. To have a covering property
In short, the covering properties of finite quasisimple groups are not necessarily uniform. It is uniform when obtained through increasing ranks, and it is uniform when obtained through base fields of increasing sizes. At least with the techniques in this paper, we cannot combine the two uniformities into one. So we must use the double covering properties.
Let G be a group with the symmetric double covering property for some parameters, and let
For any normal subgroup N, G has the symmetric double covering property for the same parameters mod N.
Any quotient group of G has the symmetric double covering property for the same parameters.
The group
As a result of (ii) and (iii), any ultraproduct
Proof.
Statements (i), (ii) and (iv) are straightforward.
To see (iii), let
For any element
So we can find
Since the above identity is true for all
So we have proven (iii). ∎
Let
Proof.
For any integer D, and for the constant
Let
will have the symmetric double covering property
Corollary 66 (Quasirandomness implies a nice covering property mod cosocle)
For any integer D, and any constant c, we can find integers
Proof.
Let
Let N be the cosocle of G. Then
Let
Proof.
Let
For any integer D, and for the constant c, we can find
Let
Since
Proof of Theorem 5.
For any integer D, let
Let
Since
8 Applications
8.1 Triangles in a quasirandom group
A quasirandom group usually contains many patterns. For example, Gowers has shown the following result:
Theorem 68 (Gowers [7, Theorem 5.1])
Pick any
Morally, if we define an x-pair to be a set
Now given a q.u.p. class, we can obtain minimally almost periodic groups via ultraproducts of sequences of increasingly quasirandom groups. Then by applying ergodic theory on the ultraproduct, more patterns similar to that of Theorem 68 might emerge. It is proven by Bergelson, Robertson and Zorin-Kranich [2] that, for a quasirandom group G in a q.u.p class, any large enough subset of
Let g be an element of a group G. Then a g-triangle is the set
Theorem 70 (Bergelson, Robertson and Zorin-Kranich [2, Theorem 1.12])
Let G be contained in a q.u.p. class. For any
8.2 Self-bohrifying groups
The application in this section is related to topological groups. We shall treat all groups in previous sections as discrete groups.
A Bohr compactification of a topological group G is a continuous homomorphism
The Bohr compactification exists for any group by the work of Holm [10]. It is obviously unique up to a unique isomorphism.
Clearly, a discrete group is minimally almost periodic iff it has trivial Bohr compactification. Note that for a discrete group, any abstract homomorphism from it to another topological group is automatically continuous.
A topological group G is said to be self-bohrifying if its Bohr compactification bG is the same abstract group as G, but with a compact topology.
By the results and techniques of this paper, one can find many examples of self-bohrifying groups. In particular, we have the following theorem.
Let n be a positive integer. Let
Let
We will prove Theorem 74 by first showing that
Let
Their sum is the group
Their reduced product is the group
Lemma 77 (Hart and Kunen [9, Lemma 3.8])
Let
Proof of Theorem 74.
All 2-quasirandom groups are perfect. So it is enough to show that the reduced product of
For any integer D, let
Let
Since
Now by Proposition 64, covering properties are preserved by arbitrary products and quotients. So
Since
Acknowledgements
I would like to thank Professor Terence Tao for introducing me to this area and for his patient guidance. I would also like to thank Professor Richard Schwartz, Professor Vitaly Bergelson, Professor Emmanuel Breuillard and Professor Nikolay Nikolov for their helpful inputs, and thank Professor László Pyber for his helpful inputs and for pointing me to a number of very useful references.
References
[1] Babai L., Goodman A. J. and Pyber L., Groups without faithful transitive permutation: Representations of small degree, J. Algebra 195 (1997), 1–29. 10.1006/jabr.1997.7042Search in Google Scholar
[2] Bergelson V., Robertson D. and Zorin-Kranich P., Triangles in Cartesian squares of quasirandom groups, preprint 2014, http://arxiv.org/abs/1410.5385. 10.1017/S0963548316000250Search in Google Scholar
[3] Bergelson V. and Tao T., Multiple recurrence in quasirandom groups, Geom. Funct. Anal. 24 (2014), no. 1, 1–48. 10.1007/s00039-014-0252-0Search in Google Scholar
[4]
Brenner J. L.,
Covering theorems for FINASIGs. VlII: Almost all conjugacy classes in
[5] Cheeger J. and Ebin D. G., Comparison Theorems in Riemannian Geometry, American Mathematical Society, Providence, 1975. Search in Google Scholar
[6] Collins M., On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007), no. 4, 411–423. 10.1515/JGT.2007.032Search in Google Scholar
[7] Gowers W. T., Quasirandom groups, Combin. Probab. Comput. 17 (2008), no. 3, 363–387. 10.1017/S0963548307008826Search in Google Scholar
[8] Grove L. C., Classical Groups and Geometric Algebra, American Mathematical Society, Providence, 2002. 10.1090/gsm/039Search in Google Scholar
[9] Hart J. E. and Kunen K., Bohr compactifications of non-abelian groups, Topology Proc. 26 (2002), no. 2, 593–626. Search in Google Scholar
[10] Holm P., On the Bohr compactification, Math. Ann. 156 (1964), 34–46. 10.1007/BF01359979Search in Google Scholar
[11] Holt D. and Plesken W., Perfect Groups, Clarendon Press, Oxford, 1989. Search in Google Scholar
[12] Kassabov M., Lubotzky A. and Nikolov N., Finite simple groups as expanders, Proc. Natl. Acad. Sci. USA 103 (2006), no. 16, 6116–6119. 10.1073/pnas.0510337103Search in Google Scholar PubMed PubMed Central
[13] von Neumann J. and Wigner E. P., Minimally almost periodic groups, Ann. of Math. (2) 41 (1940), no. 4, 746–750. 10.1007/978-3-662-02781-3_23Search in Google Scholar
[14] Sepanski M. R., Compact Lie Group, Springer Sci. Bus. Media 235, Springer, New York, 2007. 10.1007/978-0-387-49158-5Search in Google Scholar
[15] Stolz A. and Thom A., On the lattice of normal subgroups in ultraproducts of compact simple groups, Proc. Lond. Math. Soc. (3) 108 (2013), no. 1, 73–102. 10.1112/plms/pdt027Search in Google Scholar
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- The Thompson chain of subgroups of\break the Conway~group Co1 and complete graphs on ${n}$~vertices
- The influence of maximal quotient groups on the normalizer conjecture of integral group rings
- On the skew-morphisms of dihedral groups
- Minimal permutation representations of semidirect products of groups
- Triangle-free commuting conjugacy class graphs
- Upper bounds on the number of conjugacy classes in unitriangular groups
- Character degree graphs of solvable groups with diameter three
-
Feit numbers and
- Ultraproducts of quasirandom groups with small cosocles
Articles in the same Issue
- Frontmatter
- The Thompson chain of subgroups of\break the Conway~group Co1 and complete graphs on ${n}$~vertices
- The influence of maximal quotient groups on the normalizer conjecture of integral group rings
- On the skew-morphisms of dihedral groups
- Minimal permutation representations of semidirect products of groups
- Triangle-free commuting conjugacy class graphs
- Upper bounds on the number of conjugacy classes in unitriangular groups
- Character degree graphs of solvable groups with diameter three
-
Feit numbers and
- Ultraproducts of quasirandom groups with small cosocles
