Expansions and Quadraticity for Groups

First year students learn that the Taylor expansion Z carries functions into power series, and that it has some nice algebraic properties (e.g., Z (fg) = Z (f)Z (g)). It is less well known that the same game can be played within arbitrary groups: there is a natural way to say “a Taylor expansion Z for elements of an arbitrary group G”, and a natural way to carry the algebraic properties of the Taylor expansion to this more general context. In the case of a general G “Taylor expansions” (expansions with the same good properties as Z ) may or may not exist, they may or may not separate group elements, and a further good property which is hidden in the case of Z , “quadraticity”, may or may not hold. The purpose of this expository note is to properly define all the notions in the above paragraph, to enumerate some classes of groups whose theory of expansions we either understand or wish to understand, to indicate the relationship between these notions and the notions of “finite type invariants” and “unipotent” and “Mal’cev” completions, and to point out (with references) that our generalization of “expansions” to arbitrary groups is merely the tip of an iceberg, for almost everything we say can be generalized further to “expansions for arbitrary algebraic structures”.