The Number of Sides of a Parallelogram

We define parallelograms of base a and b in a group. They appear as minimal relators in a presentation of a subgroup with generators a and b. In a Lie group they are realized as closed polygonal lines, with sides being orbits of left-invariant vector fields. We estimate the number of sides of parallelograms in a free nilpotent group and point out a relation to the rank of rational series. 1 Introduction In IR 2 a parallelogram of base a and b can be defined as a closed polygon with the minimum number of sides parallel to a and b. In that paper we also consider parallelograms defined in more general groups. In section 1. we first give some definitions and examples of parallelograms in Lie groups. These examples show the various complex situations occurring in the general case. In this paper we concentrate our attention on free nilpotent groups. This analysis will give universal properties for parallelograms. We obtain Theorem. The number of sides of a parallelogram on a free nilpotent group on two generators of order n is between n and n 2. We do not know what is the exact number of sides of parallelograms in a free nilpotent group neither how many non-equivalent parallelograms exist. We hope that an investigation of parallelograms might help understand general nilpotent groups. In particular it will be interesting to find presentations with relators of minimal size. We have chosen in this paper to recall the basic properties and constructions of free Lie algebras in order to make it self-contained. That is done in section 2. In the last section we then introduce mth-order parallelograms and prove our result. A connection with rational series is pointed out at the end of the paper. Our initial motivation to study parallelograms was the notion of curvature and holonomy of a connection for Riemannian manifolds and the generalization of those notions to sub-Riemannian geometry (see [FGR] and [BeR]). In classical differential geometry, curvature appears as the quadratic term in the asymptotic expansion of holonomy around short (four-sided) parallelograms, holonomy being the 1365–8050 c ¢ measure of the difference of the vector field by parallel translation around a closed loop. In the case of sub-Riemannian manifolds, the tangent space is naturally a nilpotent group ([BeR]) and the holonomy associated to it will be calculated using parallelograms with many sides. The analog of sectional curvatures should …