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arXiv · 2608.21465

The stable commutator length of a relator is not a one-relator group invariant

Artem Semidetnov
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Abstract

We show that, for the words $r=\mathtt{aabABabABBAbaabABBAb}$ and $r'=\mathtt{aabABabABabABBAbaBAb}$, both in the commutator subgroup of the free group on $\mathtt{a}$ and $\mathtt{b}$, the one-relator groups $\langle \mathtt{a,b}\mid r=1\rangle$ and $\langle \mathtt{a,b}\mid r'=1\rangle$ are isomorphic, whereas $r$ and $r'$ have distinct stable commutator lengths. This provides a negative answer to a question posed by Heuer and L\"oh.

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Artem Semidetnov. 2026-08-21. The stable commutator length of a relator is not a one-relator group invariant. https://arxiv.org/abs/2608.21465

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