  We define invariants of words in arbitrary groups, measuring how letters in a
word are interleaving, perfectly detecting the dimension series of a group.
These are the letter-braiding invariants. On free groups, braiding invariants
coincide with coefficients in the Magnus expansion. In contrast with such
coefficients, our invariants are defined on all groups and over any PID. They
respect products in the group and are complete with respect to the dimension
series, so serve as the coefficients of a universal multiplicative finite-type
invariant, depending functorially on the group. Letter-braiding invariants
arise out of analyzing the bar construction on a cochain model of a space with
a prescribed fundamental group. This approach specializes to simplicial
presentations of a group as well as to more geometric contexts, which we
illustrate.
  As an application, we define a Johnson style filtration and a Johnson
homomorphism on the automorphisms of any group, and use it to constrain
automorphisms of finite $p$-groups.

Group Theory;Algebraic Topology;Geometric Topology