Stable commutator length is rational in free groups - Mathematics > Group TheoryReport as inadecuate




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Abstract: For any group, there is a natural pseudo-norm on the vector space B1 ofreal group 1-boundaries, called the stable commutator length norm. This normis closely related to, and can be thought of as a relative version of, theGromov pseudo-norm on ordinary homology. We show that for a free group, theunit ball of this pseudo-norm is a rational polyhedron.It follows that stable commutator length in free groups takes on onlyrational values. Moreover every element of the commutator subgroup of a freegroup rationally bounds an injective map of a surface group.The proof of these facts yields an algorithm to compute stable commutatorlength in free groups. Using this algorithm, we answer a well-known question ofBavard in the negative, constructing explicit examples of elements in freegroups whose stable commutator length is not a half-integer.



Author: Danny Calegari

Source: https://arxiv.org/







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