SQ-universal group


In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.

History

Many classic results of combinatorial group theory, going back to 1949, are now interpreted as saying that a particular group or class of groups is SQ-universal. However the first explicit use of the term seems to be in an address given by Peter Neumann to entitled "SQ-universal groups" on 23 May 1968.

Examples of SQ-universal groups

In 1949 Graham Higman, Bernhard Neumann and Hanna Neumann proved that every countable group can be embedded in a two-generator group. Using the contemporary language of SQ-universality, this result says that F2, the free group on two generators, is SQ-universal. This is the first known example of an SQ-universal group. Many more examples are now known:
In addition much stronger versions of the Higmann-Neumann-Neumann theorem are now known. Ould Houcine has proved:

Some elementary properties of SQ-universal groups

A free group on countably many generators h1, h2,..., hn,..., say, must be embeddable in a quotient of an SQ-universal group G. If are chosen such that for all n, then they must freely generate a free subgroup of G. Hence:
Since every countable group can be embedded in a countable simple group, it is often sufficient to consider embeddings of simple groups. This observation allows us to easily prove some elementary results about SQ-universal groups, for instance:
To prove this suppose N is not SQ-universal, then there is a countable group K that cannot be embedded into a quotient group of N. Let H be any countable group, then the direct product H × K is also countable and hence can be embedded in a countable simple group S. Now, by hypothesis, G is SQ-universal so S can be embedded in a quotient group, G/M, say, of G. The second isomorphism theorem tells us:
Now and S is a simple subgroup of G/M so either:
or:
The latter cannot be true because it implies K ⊆ H × K ⊆ S ⊆ N/ contrary to our choice of K. It follows that S can be embedded in /, which by the third isomorphism theorem is isomorphic to G/MN, which is in turn isomorphic to /. Thus S has been embedded into a quotient group of G/N, and since H ⊆ S was an arbitrary countable group, it follows that G/N is SQ-universal.
Since every subgroup H of finite index in a group G contains a normal subgroup N also of finite index in G, it easily follows that:

Variants and generalizations of SQ-universality

Several variants of SQ-universality occur in the literature. The reader should be warned that terminology in this area is not yet completely stable and should read this section with this caveat in mind.
Let be a class of groups. A group G is called SQ-universal in the class if and every countable group in is isomorphic to a subgroup of a quotient of G. The following result can be proved:
Let be a class of groups. A group G is called SQ-universal for the class if every group in is isomorphic to a subgroup of a quotient of G. Note that there is no requirement that nor that any groups be countable.
The standard definition of SQ-universality is equivalent to SQ-universality both in and for the class of countable groups.
Given a countable group G, call an SQ-universal group H G-stable, if every non-trivial factor group of H contains a copy of G. Let be the class of finitely presented SQ-universal groups that are G-stable for some G then Houcine's version of the HNN theorem that can be re-stated as:
However, there are uncountably many finitely generated groups, and a countable group can only have countably many finitely generated subgroups. It is easy to see from this that:
An infinite class of groups is wrappable if given any groups there exists a simple group S and a group such that F and G can be embedded in S and S can be embedded in H. The it is easy to prove:
The motivation for the definition of wrappable class comes from results such as the Boone-Higman theorem, which states that a countable group G has soluble word problem if and only if it can be embedded in a simple group S that can be embedded in a finitely presented group F. Houcine has shown that the group F can be constructed so that it too has soluble word problem. This together with the fact that taking the direct product of two groups preserves solubility of the word problem shows that:
Other examples of wrappable classes of groups are:
The fact that a class is wrappable does not imply that any groups are SQ-universal for. It is clear, for instance, that some sort of cardinality restriction for the members of is required.
If we replace the phrase "isomorphic to a subgroup of a quotient of" with "isomorphic to a subgroup of" in the definition of "SQ-universal", we obtain the stronger concept of S-universal. The Higman Embedding Theorem can be used to prove that there is a finitely presented group that contains a copy of every finitely presented group. If is the class of all finitely presented groups with soluble word problem, then it is known that there is no uniform algorithm to solve the word problem for groups in. It follows, although the proof is not a straightforward as one might expect, that no group in can contain a copy of every group in. But it is clear that any SQ-universal group is a fortiori SQ-universal for. If we let be the class of finitely presented groups, and F2 be the free group on two generators, we can sum this up as:
The following questions are open :
While it is quite difficult to prove that F2 is SQ-universal, the fact that it is SQ-universal for the class of finite groups follows easily from these two facts:
If is a category and is a class of objects of, then the definition of SQ-universal for clearly makes sense. If is a concrete category, then the definition of SQ-universal in also makes sense. As in the group theoretic case, we use the term SQ-universal for an object that is SQ-universal both for and in the class of countable objects of.
Many embedding theorems can be restated in terms of SQ-universality. Shirshov's Theorem that a Lie algebra of finite or countable dimension can be embedded into a 2-generator Lie algebra is equivalent to the statement that the 2-generator free Lie algebra is SQ-universal. This can be proved by proving a version of the Higman, Neumann, Neumann theorem for Lie algebras. However versions of the HNN theorem can be proved for categories where there is no clear idea of a free object. For instance it can be proved that every separable topological group is isomorphic to a topological subgroup of a group having two topological generators.
A similar concept holds for free lattices. The free lattice in three generators is countably infinite. It has, as a sublattice, the free lattice in four generators, and, by induction, as a sublattice, the free lattice in a countable number of generators.