Correction: Root extraction in one-relator groups and slenderness

ABSTRACT

This very short correction notes a gap in an argument of an earlier paper, and also provides a theorem of similar flavor to the main result of that paper.

KEYWORDS:

• one-relator
• automatic continuity

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20F05
• 57M07

This article refers to:

Root extraction in one-relator groups and slenderness

I am indebted to Dawid Kielak [8] for pointing out a gap in the proof of Lemma 2.4 in [3], which was used to prove Theorems A and B of that paper. There are no known counterexamples to those results, so they may be regarded as open problems. The problem with that original argument is that if L and K are two Magnus subgroups of a one-relator group, then the expression of an element in the intersection $L\cap K$ as a word in the free generating set of L can be different from that in the free generating set for K. I’ll state and prove a result similar to Theorem B with stronger conclusion but an extra assumption (that the one-relator group includes no Baumslag-Solitar subgroups). The proof similarly utilizes root extraction.

The notation will be identical with that of the paper [3], with HEG, $\mathbf{HE}{\mathbf{G}}^m$, etc., as therein. We’ll recall the following definitions from [2].

Definition.

A group G is cm-slender (respectively lcH-slender) if every abstract group homomorphism $\varphi :H\to G$, where H is a completely metrizable (resp. locally compact Hausdorff) topological group, has open kernel.

Theorem.

Let G be a (possibly uncountable) one-relator group which has no Baumslag-Solitar subgroup. The following hold.

1. If $\varphi :\mathbf{HEG}\to G$ is an abstract homomorphism then for some $m\in \mathbb{N}$ the image $\varphi (\mathbf{HE}{\mathbf{G}}^m)$ is finite.

2. If $\varphi :H\to G$ is an abstract homomorphism, with H either a completely metrizable or locally compact Hausdorff topological group, then there is a normal open subgroup $V\le H$ with $\varphi (V)$ finite.

In particular, a torsion-free one-relator group without Baumslag-Solitar subgroups is n-slender, cm-slender, and lcH-slender.

Proof.

We will first prove claim (1) and then give the quite analogous argument for (2). Assume that $\varphi :\mathbf{HEG}\to G$ is an abstract group homomorphism, where G is a one-relator group without Baumslag-Solitar subgroups. If $G=〈X|r〉$ we let $Y\subseteq X$ be the set of generators used in the word r. The group G is isomorphic in the natural way to the free product $〈Y|r〉*F(X\setminus Y)$ where $F(X\setminus Y)$ is the free group on generators $X\setminus Y$. Then $\varphi :\mathbf{HEG}\to 〈Y|r〉*F(X\setminus Y)$, so by [6, Theorem 1.3] we know that for some $N\in \mathbb{N}$ the image $\varphi (\mathbf{HE}{\mathbf{G}}^N)$ is included into a conjugate of either $〈Y|r〉$ or $F(X\setminus Y)$. Thus without loss of generality we compose $\varphi$ with conjugation by an appropriate element in G so that $\varphi (\mathbf{HE}{\mathbf{G}}^N)\le 〈Y|r〉$ or $\varphi (\mathbf{HE}{\mathbf{G}}^N)\le F(X\setminus Y)$. In case $\varphi (\mathbf{HE}{\mathbf{G}}^N)\le F(X\setminus Y)$, since free groups are n-slender [5, Corollary 3.7], we can select $m>N$ such that $\mathbf{HE}{\mathbf{G}}^m\le \ker (\varphi \upharpoonright \mathbf{HE}{\mathbf{G}}^N)\le \ker (\varphi )$, so $\varphi (\mathbf{HE}{\mathbf{G}}^m)$ is trivial, hence finite.

Suppose now that $\varphi (\mathbf{HE}{\mathbf{G}}^N)\le 〈Y|r〉$. In case $J=〈Y|r〉$ has torsion we know it is hyperbolic (by the Spelling Theorem of Newman [10]). Then there exists some m > N such that $\varphi (\mathbf{HE}{\mathbf{G}}^m)$ is finite [1, Theorem B]. Therefore we may assume that J is torsion-free. As G does not have Baumslag-Solitar subgroups, we know that J is commutative transitive [7, Theorem 1.3] and so has unique root extraction (i.e. if $s_0^t=s_1^t$, t > 0, then s₀ = s₁). Letting p be a prime greater than the length of the relator r, we have that for each nontrivial $s\in J$ there is some $n_s\in \mathbb{N}$ such that the equation $x^{p^{n_s}}=s$ has no solution in J [10, Theorem 1]. Then by unique root extraction we have that for nontrivial $s\in 〈Y|r〉$, the set $\{x\in J:(\exists k\in \mathbb{N})x^{p^k}=s\}$ has cardinality at most n_s . Then in the terminology of [2] the group J has finite p-antecedents, and as J is countable and torsion-free, we know that J is n-slender [2, Theorems A, B(c)]. Then there exists some m > N such that $\varphi \upharpoonright \mathbf{HE}{\mathbf{G}}^m$ is trivial and we have considered the last case for (1).

Now we’ll prove (2). Letting $\varphi :H\to G$ with H completely metrizable (respectively locally compact Hausdorff) and $G=〈Y|r〉*F(X\setminus Y)$, with Y finite, we have that either $\ker (\varphi )$ is open or $\varphi (H)$ lies entirely in a conjugate of $〈Y|r〉$ or of $F(X\setminus Y)$ by [11] (resp. [9]). In case $\ker (\varphi )$ is open we are already done. Else we conjugate appropriately so that without loss of generality either $\varphi (H)\le 〈Y|r〉$ or $\varphi (H)\le F(X\setminus Y)$. If $\varphi (H)\le F(X\setminus Y)$ then since free groups are cm-slender [4] (resp. lcH-slender, also [4]) we see again that $\ker (\varphi )$ is open. We are left with the case where $\varphi (H)\le 〈Y|r〉$. If the group $J=〈Y|r〉$ has torsion then it is hyperbolic and by [1, Theorem A] there is an open normal subgroup $V\le H$ such that $\varphi (V)$ is finite. If J is torsion-free, then as in (1) J has finite p-antecedents and we have J is cm-slender and lcH-slender [2, Theorems A, B(c)]. Then $\ker (\varphi )$ is open and the last case for (2) is complete.□

It should be noted that Baumslag-Solitar groups are themselves known to be n-slender, cm-slender, and lcH-slender [2, Theorems A, B(i)]. Finally, we point out that a positive answer to the following question allows one to remove the requirement that the group has no Baumslag-Solitar subgroups.

Question 1. If G is a torsion-free one-relator group and p is a prime number greater than the length of the relator of G, then does G have finite p-antecedents?

Funding

Acknowledgments

The author thanks the referee for helpful suggestions.

Additional information

Funding

This work was supported by the Additional Funding Programme for Mathematical Sciences, delivered by EPSRC (EP/V521917/1) and the Heilbronn Institute for Mathematical Research.