The stable commutator length of an element 
in the commutator subgroup
of a group 
is
 |
where 
is the commutator length. The limit
exists because commutator length is subadditive
(Calegari 2009a).
For an element 
in the commutator subgroup of a free
group, 
(Duncan and Howie 1991). Stable commutator length takes values in the rational
numbers and is computable in free groups (Calegari
2009b).
Heuer and Löh (2022) asked whether the stable commutator length of a defining relation is determined by the corresponding one-relator
group. Semidetnov (2026) gave the two words 
and 
in the free
group on generators 
and 
, where 
and 
. The associated one-relator
groups are isomorphic, but 
and 
, giving a negative answer. Semidetnov (2026) credits
Claude Opus 5 with assisting in the design and orchestration of the exhaustive search
for the counterexample.
