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

Abstract

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

3 pages