A new characterization of braid groups and rings of invariants for symmetric automorphism groups (Q1358219)

Let \(F(n)=\langle x_1,\dots,x_n\rangle\) be a free group. An automorphism \(\alpha\) of \(F(n)\) is symmetric if \(\alpha(x_i)=w_ix_jw_i^{-1}\) for \(i=1,\dots,n\), where \(i\to j\) is a permutation and \(w_i\in F(n)\). The group of symmetric automorphisms is denoted by \(\widehat H(n)\). \(\widehat H(n)\) contains the braid groups \(B_n\). In an earlier paper we described the action of \(\widehat H(n)\), as automorphisms of a polynomial algebra \(Z\). In this paper we give conditions defining the image of \(B_n\) as such automorphisms. This involves understanding the stabilisers of certain elements of \(Z\), which are themselves traces of matrices over \(Z\). We then study geometric questions concerning closed braids, in particular Markov's second move. We show that one can naturally lift this action to one on a non-commutative ring. We also determine the ring of invariants for this (non-linear) action of \(\widehat H(n)\) on \(Z\) and describe more elements of \(Z\) invariant under the action of \(B_n\).