A family \(\mathfrak{V}\) of topological groups is said to be a variety if it is closed under products, subgroups and topological group quotients. A 43-year old problem of A. V. Arhangel’skiĭ states the following:
Question. Let \(\mathfrak{V}\) be the smallest variety of topological groups containing all free topological groups on metric spaces. Is \(\mathfrak{V}\) exactly the variety of all topological groups?
In this outstanding paper, the author gives a negative answer to this problem, assuming the existence of certain real-valued measurable cardinals. The author drew inspiration from the following notion introduced in a joint manuscript by S. A. Morris et al. [“The variety generated by free topological groups on metrizable spaces”, Manuscript]:
Definition. A topological group \(G\) is \(g\)-sequential if either one of the following equivalent properties hold:

1.

for any topological group \(H\), any sequentially continuous homomorphism \(f: G \to H\) is continuous;

2.

\(G\) admits no strictly finer group topology with the same convergent sequences; and

3.

\(G\) is isomorphic to a quotient group of the free topological group of a metrizable space.

By proposing to consider the variety \(\mathfrak{V}\) generated by all \(g\)-sequential topological groups, the author studies the validity of the following statement:
(G) Every topological group is isomorphic to a subgroup of a \(g\)-sequential topological group.
When (G) holds true, then this implies a positive answer to the problem of A. V. Arhangel’skiĭ. To this end, the author introduces the auxiliary concept of a \(g\)-sequential cardinal:
Definition 1.3. A subadditive measure on a set \(E\) is a function \(p: \mathcal{P}(E) \to [0,1]\) from the power set \(\mathcal{P}(E)\) to the unit interval satisfying the following:

1.

\(p(A \cup B) \leq p(A) + p(B)\) holds for all \(A,B \in \mathcal{P}(E)\);

2.

for any decreasing sequence \(A_1 \supset A_2 \supset \dots\) of elements of \(\mathcal{P}(E)\) with empty intersection one has \(\lim p(A_n) = 0\). A cardinal \(\tau = \mathrm{Card}(E)\) is \(g\)-sequential if there exists a subadditive measure \(p\) on \(E\) such that \(p(E) = 1\) and \(p(F) = 0\) for all finite subsets \(F \subset E\).

The author notes that the existence of a real-valued measurable cardinal implies the existence of a \(g\)-sequential cardinal. The following result of the author thus proves that the (G) is incompatible with the existence of real-valued cardinals:
Theorem 2.13. Assume there exist \(g\)-sequential cardinals. Then, the variety \(\mathfrak{V}\) generated by free topological groups of metrizable spaces is a proper subclass of all topological groups. In particular, if \(\mathrm{Card}(E)\) is a \(g\)-sequential cardinal, the topological group \(\mathrm{Sym}(E)\) does not belong to \(\mathfrak{V}\).
For this reason, the author proposes the following additional statement:
(M) if there are no \(g\)-sequential cardinals, then the problem by Arhangel’skiĭ has a positive solution.
In order to prove this statement, the author proposes to establish the following:
(U) For every Tychonoff space \(X\), the free topological group \(F(X)\) admits a topologically faithful unitary representation which is isomorphic to a subgroup of the unitary group \(U(H)\) of some (non-separable) Hilbert space.
The author then establishes the following result, which proves that if (U) holds, then (M) is true as well:
Theorem 4.10. Let \(\tau\) be a cardinal and \(G\) be a topological group satisfying one of the following conditions:

(a)

\(G\) is the group \(\operatorname{Aut} I^\tau\) of all self-homeomorphisms of a Hilbert cube \(I^\tau\),

(b)

\(G\) is the group \(\mathrm{Sym}(E)\) of all permutations of a set \(E\); or

(c)

\(G\) is the unitary group \(U(H)\) of a Hilbert space \(H\).

Then:

1.

the group \(G\) has countable functional tightness if and only if its weight is non Ulam-measurable.

2.

the group \(G\) is \(g\)-sequential if and only if its weight is not a \(g\)-sequential cardinal.

The author also explores other properties of \(g\)-sequential groups and even provides a full characterization for locally compact groups to satisfy the property:
Theorem 3.14. A locally compact group \(G\) is \(g\)-sequential if and only if the local weight of \(G\) is not a \(g\)-sequential cardinal.
We strongly suggest the interested reader to read the manuscript in its entirety for further details, more historical background and an important collection of results which are of interest.