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We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling $1$
–chains.
Given groups $A$
and $B$
, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$
not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$
and $B$
). Other similar problems are also considered.
