Computing commutator length in free groups
We study commutator length in free groups. (By a commutator lengthcl(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 F2 (Thm. 1). Moreover, for every element z ε F′2 and for any natural m, the following estimate derives:cl(zm) ≥ (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 PL. The algebra of pairs arises naturally in proving Theorem 2 and enjoys a number of interesting properties. We state that in a free group F2k with free generators a1, b1, ..., ak, bk, k εN, every natural m satisfiescl(([a1, b1] ... [ak, bk])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 F2 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.
KeywordsArbitrary Element Orientable Surface Free Generator Zero Divisor Cyclic Shift
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