An exotic presentation of ZxZ

BARMAK, JONATHAN ARIEL

Seminario; SEMINARIO A DISTANCIA DE TEORÍA GEOMÉTRICA DE GRUPOS; 2022

The Generalized Andrews-Curtis Conjecture states that if K and L are simple homotopy equivalent 2-dimensional complexes, then they are equivalent via a 3-deformation (expansions and collapses involving only complexes of dimension at most 3). There is an equivalent formulation in terms of group presentations connected via a sequence of elementary transformations. One of them is the stabilization transformation which consists of adding a new generator x and a new relator r=x. We will construct an example of two presentations (both with two generators and two relators) which are connected by elementary transformations, and we will show that the stabilization move cannot be avoided. Examples of this kind are known, but this is the first which occurs in non-minimal Euler characteristic, where most invariants fail to distinguish such situations. This example is an application of the Winding invariant, a map from F_2’ to the ring of Laurent polynomials in two variables.