# Computing commutator length in free groups

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## Abstract

We study commutator length in free groups. (By a commutator length*cl*(g) of an element g in a derived subgroup G′ of a group G we mean the least natural number k such that g is a product of k commutators.) A purely algebraic algorithm is constructed for computing commutator length in a free group F_{2} (Thm. 1). Moreover, for every element z ε F′_{2} and for any natural m, the following estimate derives:*cl*(z^{m}) ≥ (ms(z) + 6)/12, where s(z) is a nonnegative number defined by an element z (Thm. 2). This estimate is used to compute commutator length of some particular elements. By analogy with the concept of width of a derived subgroup known in group theory, we define the concept of width of a derived subalgebra. The width of a derived subalgebra is computed for an algebra P of pairs, and also for its corresponding Lie algebra P_{L}. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F_{2k} with free generators a_{1}, b_{1}, ..., a_{k}, b_{k}, k ε*N*, every natural m satisfies*cl*(([a_{1}, b_{1}] ... [a_{k}, b_{k}])^{m})=[(2 − m)/2] + mk. For k=1, this entails a known result of Culler. The notion of a growth function as applied to a finitely generated group G is well known. Associated with a derived subgroup of F_{2} is some series depending on two variables which bears information not only on the number of elements of prescribed length but also on the number of elements of prescribed commutator length. A number of open questions are formulated.

### Keywords

Arbitrary Element Orientable Surface Free Generator Zero Divisor Cyclic Shift## Preview

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