TEST ELEMENTS FOR ENDOMORPHISMS OF FREE GROUPS AND ALGEBRAS

There are two well-known approaches to recognizing automorphisms of a free group, i.e., to distinguishing automorphisms from non-automorphis ms. The first one is the ''inverse function theorem'' of Birman. The s econd one, the ''test element'' approach, was originated by Nielsen fo r the free group of rank 2 and then extended to free groups of arbitra ry finite rank by Zieschang, Rosenberger and others. In this note, we establish a direct connection between these two approaches: we associa te a special matrix with any element of a free group, and show that an automorphism can be distinguished from a non-automorphism in terms of invertibility of such a matrix associated with a particular single el ement. Similar results hold for free associative and Lie algebras.