The winding invariant

Abstract

Every element w in the commutator subgroup of the free group F2 of rank 2 determines a closed curve in the grid Z × R ∪ R × Z ⊆ R 2 . The winding numbers of this curve around the centers of the squares in the grid are the coefficients of a Laurent polynomial Pw in two variables. This basic definition is related to well-known ideas in combinatorial group theory. We use this invariant to study equations over F2 and over the free metabelian group of rank 2. We give a number of applications of algebraic, geometric and combinatorial flavor. CONTENTS 2010

Invariants for metabelian groups of prime power exponent, colorings and stairs

“…Algebraic tools have been used to study Engel identities and the nilpotence class of M (2, n) in works by Bachmuth-Heilbronn-Mochizuki [7] and Dark-Newell [14]. Here we use instead combinatorial ideas based on a construction introduced in [9], called the winding invariant. To each word w in the derived subgroup F ′ 2 of the free group F 2 = F (x, y) of rank 2, we associate a Laurent polynomial P w ∈ Z[X ±1 , Y ±1 ].…”

“…Both notions are related by the identity em = em(x −1 , y −1 ) −1 , so the definitions of m-Engel group coincide. cases in which n is prime, the order of M (d, n) is known in the following cases: (d, n) ∈ {(2, 4), (3,4), (4,4), (5,4), (2,8), (3,8), (4,8), (2,9), (3,9)} (computed by Gupta-Tobin [20], Hermanns [27] and Newman [39]). The order of M (2, 16) is known to be smaller than or equal to 2 376 ( [39]).…”

“…We start by recalling the constructions and results of [9] that we will need. We will also recall some of the proofs for the sake of completeness and in order to make clear subsequent ideas.…”

“…In [9] we proved that the area invariant A : F ′ 2 → Z defined by A(z) = P z (1, 1), maps a product of nth powers in F 2 to a multiple of n/2 when n is even and to a multiple of n when n is odd. In particular a product of 4th powers has even area.…”

“…. (m j−1 − 1) where the m i are monomials (see also [9,Proposition 23]). If P ∈ Z[X ±1 , Y ±1 ] and m is a monomial, Ω(P ) and Ω(mP ) have the same coordinates up to a shift and change of sign.…”