# Automatic continuity (Part 3)

In this installment I’ll present some newer automatic continuity results in the spirit of Specker’s theorem.  Recall that a group $G$ is cm-slender (respectively lcH-slender, n-slender) if every abstract group homomorphism from a completely metrizable topological group (resp. locally compact Hausdorff topological group, the Hawaiian earring group) to $G$ has open kernel (see Part 2).  A group satisfying any of these slenderness notions must be both torsion-free and not contain $\mathbb{Q}$ as a subgroup.

On initially understanding Higman’s proof that free groups are n-slender, one notices that the proof can be applied to a wider range of groups.  The proof utilizes the fact that the natural length function on a free group satisfies certain nice properties, together with a diagonalization argument.  We’ll say a function $L: G \rightarrow \omega$ is a length function provided $L(1_G) = 0$, $L(g) = L(g^{-1})$ and $L(gh) \leq L(g) + L(h)$.  A length function $L$ is uniformly monotone (abbrev. u.m.) provided there exists some $k\in \omega$ for which $g\in G\setminus \{1_G\}$ implies $L(g^k) \geq L(g) +1$.  Clearly a u.m. length function cannot exist if $G$ has torsion or more generally if $G$ has a nontrivial element $g$ which has infinitely many roots (i.e. for infinitely many $n\in \omega$ there exists $h_n$ for which $h_n^n = g$).  In the case of a free group one can use constant $k = 2$.  The existence of a u.m. length function implies that a group is n-slender by Higman’s argument.  Dudley (in Continuity of homomorphisms, 1961) used a different sort of length function (which he called a norm) to prove slenderness  results about free (abelian) groups.

One can show that torsion-free word hyperbolic groups always have a u.m. length function (see Torsion-free word hyperbolic groups are noncommutatively slender, 2016), and are therefore n-slender.  A u.m. length function argument also shows that the class of n-slender groups is closed under taking graph products (not just free products or direct sums).

By taking a still more general approach one can prove many more automatic continuity theorems.  In the following three paragraphs I’ll describe some results with Greg Conner in A note on automatic continuity, 2019.  Given a group $G$, subset $S \subseteq G$, and $j \in \omega \setminus \{0\}$ we let $\sqrt[j]{S} = \{g\in G \mid g^j\in S\}$.  A limiting sequence pair for a group $G$ is a pair of sequences $F_1 \subseteq F_2\subseteq \cdots$ and $k_1, k_2, \ldots$ with $F_n \subseteq G$ and $k_n \in \omega$ such that for every natural number $n$:

• $(\forall g \in G)(\exists m\in \mathbb{N})[gF_n \subseteq F_m]$
• $\sqrt[k_n]{F_n} =\{1_G\}$
• $\sqrt[k_m]{F_n} \subseteq F_n, ~ \forall m \leq n$

A group with a limiting sequence pair is cm-, lcH-, and n-slender (see Theorem A).  This gives a plethora of automatic continuity results since each of the following has a limiting sequence pair (see Theorem B):

• groups with a Dudley norm (see above in this post)
• groups with uniformly monotone length function
• countable torsion-free groups with finite roots
• $\mathbb{Z}[\frac{1}{m}]$
• Baumslag-Solitar groups
• Thompson’s group $F$

A few other such groups could be mentioned, but this list gives a sense of the breadth of the class of slender groups and of the applicability of the notion of limiting sequence pair.  In this paper, it is also shown that the class of cm-slender groups, the class of lcH-slender groups, and the class of n-slender groups are closed under taking graph products.

A class of groups missing from this list is the torsion-free one-relator groups.  By strengthening a lemma due to Newman (Some results on one-relator groups, 1968, Lemma 2) and inductively applying a result of Nakamura (Atomic properties of the Hawaiian earring group for HNN extensions, 2015) I show that such groups are n-slender (Root extraction in one-relator groups and slenderness, 2018).  This proves a conjecture in Nakamura’s paper.

Some other groups which are reasonably close to being free groups are also slender.  For example, if $G$ is a group in which every countable subgroup is free (so-called $\aleph_1$-free) then $G$ is cm-, lcH- and n-slender (see Automatic continuity of $\aleph_1$-free groups, to appear).  With Ilya Kazachkov we also show that various types of braid groups are slender (On preservation of automatic continuity, 2019).  More recently with Oleg Bogopolski (An atomic property for acylindrically hyperbolic groups) we prove that for any abstract homomorphism $\phi: H \rightarrow G$ with $G$ an acylindrically hyperbolic group and $H$ either completely metrizable, locally compact Hausdorff, or the Hawaiian earring group there exists a neighborhood of identity of the domain which maps into the set of elliptic elements.  From this, one can deduce that a torsion-free relatively hyperbolic group over a collection of n-slender groups is also n-slender, and similarly for a cm- or lcH-slender group provided the group is of cardinality less than $2^{\aleph_0}$.