Powers of Commutators as Products of Squares
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منابع مشابه
Powers of a product of commutators as products of squares
We prove that for any odd integer N and any integer n > 0, the Nth power of a product of n commutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integers n, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a pr...
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A classification of the ways in which an element of a free group can be expressed as a product of commutators or as a product of squares is given. This is then applied to some particular classes of elements. Finally, a question about expressing a commutator as a product of squares is addressed.
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Waring’s problem asks whether every natural number is a sum of g(k) kth powers (where g is a suitable function). This was solved affirmatively by Hilbert in 1909. Optimizing g(k) has been a central problem in additive number theory ever since (see [Na] for more detail and background). Recently there has been considerable interest in group theoretic analogues of this phenomenon, where the aim is...
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