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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:
I am indebted to Dawid Kielak [Citation8] for pointing out a gap in the proof of Lemma 2.4 in [Citation3], 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 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 [Citation3], with HEG, , etc., as therein. We’ll recall the following definitions from [Citation2].
Definition.
A group G is cm-slender (respectively lcH-slender) if every abstract group homomorphism , 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.
If
is an abstract homomorphism then for some
the image
is finite.
If
is an abstract homomorphism, with H either a completely metrizable or locally compact Hausdorff topological group, then there is a normal open subgroup
with
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 is an abstract group homomorphism, where G is a one-relator group without Baumslag-Solitar subgroups. If
we let
be the set of generators used in the word r. The group G is isomorphic in the natural way to the free product
where
is the free group on generators
. Then
, so by [6, Theorem 1.3] we know that for some
the image
is included into a conjugate of either
or
. Thus without loss of generality we compose
with conjugation by an appropriate element in G so that
or
. In case
, since free groups are n-slender [5, Corollary 3.7], we can select
such that
, so
is trivial, hence finite.
Suppose now that . In case
has torsion we know it is hyperbolic (by the Spelling Theorem of Newman [Citation10]). Then there exists some m > N such that
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
, t > 0, then s0 = s1). Letting p be a prime greater than the length of the relator r, we have that for each nontrivial
there is some
such that the equation
has no solution in J [10, Theorem 1]. Then by unique root extraction we have that for nontrivial
, the set
has cardinality at most ns
. Then in the terminology of [Citation2] 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
is trivial and we have considered the last case for (1).
Now we’ll prove (2). Letting with H completely metrizable (respectively locally compact Hausdorff) and
, with Y finite, we have that either
is open or
lies entirely in a conjugate of
or of
by [Citation11] (resp. [Citation9]). In case
is open we are already done. Else we conjugate appropriately so that without loss of generality either
or
. If
then since free groups are cm-slender [Citation4] (resp. lcH-slender, also [Citation4]) we see again that
is open. We are left with the case where
. If the group
has torsion then it is hyperbolic and by [1, Theorem A] there is an open normal subgroup
such that
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
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?
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Acknowledgments
The author thanks the referee for helpful suggestions.
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References
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