Automatic continuity (Part 2)

So far we have seen some examples of discontinuous homomorphisms between topological groups. In this contrasting post we will review some automatic continuity theorems. Our focus in this and the subsequent post will be on automatic continuity results of a particular strong flavor: maps with open kernels. For a very nice survey which includes other forms of automatic continuity (including measure theoretic) one may consult Rosendal’s Automatic continuity of group homomorphisms, 2009.

One classical motivating theorem, which provided a pattern that has been used time and again in later results, is due to Ernst Specker (see Additive Gruppen von folgen Ganzer Zahlen, 1950):

Theorem For each homomorphism $\phi: \prod_{\omega} \mathbb{Z} \rightarrow \mathbb{Z}$ there exists some $n\in \omega$ such that $\phi = \phi \circ p_n$ , where $p_n: \prod_{\omega}\mathbb{Z} \rightarrow \prod_{m=0}^{n-1}\mathbb{Z} \times (0)_{m=n}^{\infty}$ is projection to the first $n$ coordinates.

The group $\prod_{\omega}\mathbb{Z}$ can be considered a topological group by giving each coordinate group $\mathbb{Z}$ a discrete topology and the group $\prod_{\omega}\mathbb{Z}$ the product (Tychonov) topology. The theorem can be equivalently read as: any homomorphism from the topological group $\prod_{\omega}\mathbb{Z}$ to the discrete group $\mathbb{Z}$ has open kernel (and is therefore continuous). One nice corollary to Specker’s theorem is the fact that the group $\prod_{\omega} \mathbb{Z}$ is not free abelian (originally this was shown by Reinhold Baer in Abelian groups without elements of finite order, 1937). To derive the corollary you note that $\prod_{\omega}\mathbb{Z}$ is of cardinality $2^{\aleph_0}$ so that if it were free abelian it would be of rank $2^{\aleph_0}$ (i.e. a free abelian generating set would have this cardinality). This implies that the number of elements in $Hom(\prod_{\omega}\mathbb{Z}, \mathbb{Z})$ is $2^{2^{\aleph_0}}$ , but Specker’s theorem easily implies that $Hom(\prod_{\omega}\mathbb{Z}, \mathbb{Z})$ is countable.

Specker’s theorem can be proved by contradiction using a nice diagonalization argument. A generalization of Specker’s result was proved several years later by Sasiada (Proof that every countable and reduced torsion-free abelian group is slender, 1959):

Theorem Let $A$ be an abelian group of cardinality less than $2^{\aleph_0}$ . Then every homomorphism $\phi: \prod_{\omega}\mathbb{Z} \rightarrow A$ has open kernel if and only if $A$ contains no torsion and no subgroup isomorphic to $\mathbb{Q}$ .

The strongest result in this line is due to Nunke (see Slender groups, 1961): if $A$ is abelian then every homomorphism from $\prod_{\omega}\mathbb{Z}$ to $A$ has open kernel if and only if $A$ does not contain torsion or include a subgroup isomorphic to $\mathbb{Q}$ , $\prod_{\omega}\mathbb{Z}$ or the p-adic integers $J_p$ for any prime $p$ . Abelian groups satisfying either of these equivalent properties are called slender.

Leaving abelian groups, one can consider automatic continuity with regard to fundamental groups. The prototypical space with wild fundamental group is the Hawaiian earring. This space can be seen as a shrinking wedge of countably infinitely many circles. More formally we let $C_{p, r}$ denote the circle in the plane $\mathbb{R}^2$ centered at point $p$ of radius $r$ . The Hawaiian earring is the subspace $E = \bigcup_{m\in \omega} C_{(\frac{1}{m+1} 0), \frac{1}{m+1}}$ . Although this space superficially looks like a countably infinite bouquet of circles, its fundamental group (which we denote $HEG$ ) is not free and is in fact uncountable. The space $E$ can be retracted to the wedge of the outer $n$ circles $\bigcup_{m = 0}^{n-1}C_{(\frac{1}{m+1}, 0), \frac{1}{m+1}}$ by mapping all points in $E$ which are not on this outer wedge to the point $(0, 0)$ . This continuous function induces a homomorphic retraction $p_n$ from the group $HEG$ to a subgroup $HEG_n$ which is isomorphic to a free group of rank $n$ . There is a complementary subgroup $HEG^n$ which is the image of the retraction given by mapping the outer $n$ circles to the point $(0, 0)$ . One has the free product decomposition $HEG = HEG_n * HEG^n$ .

The group $HEG$ can be understood as a nonabelian fundamental group analogue of $\prod_{\omega}\mathbb{Z}$ . The infinite torus $\prod_{\omega}S^1$ has fundamental group isomorphic to $\prod_{\omega}\mathbb{Z}$ , and the homomorphic retractions $p_n: \prod_{\omega}\mathbb{Z} \rightarrow \prod_{m=0}^{n-1}\mathbb{Z} \times (0)_{m=n}^{\infty}$ can be induced by topological retractions of $\prod_{\omega}S^1$ to $\prod_{m = 0}^{n-1}S^1 \times (x)_{m=n}^{\infty}$ where $x\in S^1$ is some distinguished point.

We give some definitions which hearken back to Specker’s theorem: a group $G$ is noncommutatively slender (abbrev. n-slender) if for every homomorphism $\phi: HEG \rightarrow G$ there exists some $n\in \omega$ for which $\phi = \phi \circ p_n$ (see Eda’s Free $\sigma$ -products and noncommutatively slender groups, 1992). Alternatively $G$ is n-slender if for each homomorphism from $HEG$ to $G$ the kernel includes $HEG^n$ for some $n\in \omega$ . We’ll say a group $G$ is completely metrizable slender (abbrev. cm-slender) if any abstract group homomorphism from a completely metrizable topological group to $G$ has kernel which is open. Analogously define a group $G$ to be locally compact Hausdorff slender (abbrev. lcH-slender) if every abstract group homomorphism whose domain is a locally compact Hausdorff topological group and whose codomain is $G$ must have open kernel.

Graham Higman proved that free groups are n-slender, though he did not use this precise terminology (see Theorem 1 of Unrestricted free products and varieties of topological groups, 1952). By using a modification of Higman’s proof, Dudley showed that free (abelian) groups are n-, cm-, and lcH-slender (Continuity of homomorphisms, 1962). I’ll finish this post by providing a mini-catalogue of related results:

If $G = *_{i\in I} H_i$ then for any abstract homomorphism $\phi: H \rightarrow G$ with $H$ locally compact either $\ker(\phi)$ is open or $\phi(H) \leq g^{-1} H_i g$ for some $i\in I$ and $g\in G$ (Morris and Nickolas, Locally compact group topologies on an algebraic free product of groups, 1976, Theorem 3). This immediately implies that the class of lcH-slender groups is closed under taking free products.
Same as Morris and Nickolas, except $H$ is a completely metrizable topological group (Slutsky, Automatic continuity for homomorphisms into free products, 2013, Theorem 5.6). As a result, the cm-slender groups are closed under free products.
If $G = *_{i\in I} H_i$ and $\phi: HEG \rightarrow G$ is an abstract homomorphism then for some $n\in \omega$ , $i\in I$ and $g\in G$ we have $\phi(HEG^n) \leq g^{-1} H_i g$ (Eda, Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces, 2011, Theorem 1.3). In particular the class of n-slender groups is closed under free products; this latter fact, as well as the closure of the class of n-slender groups under direct sums, was proved earlier by Eda (Free $\sigma$ -products and noncommutatively slender groups, 1992, Theorem 3.6). In this latter paper it was also shown that the abelian n-slender groups are precisely the slender groups (Theorem 3.3).
Baumslag-Solitar groups are n-slender (Nakamura, Atomic properties of the Hawaiian earring group for HNN extensions, 2015).