On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

A composition formula in the rank two free group

Using the fundamental group of a punctured torus, a free group of rank two, and the fact that the natural eipmorphism from onto has as kernel the group of inner automorphisms of , we describe representatives of the conjugacy classes of generating pairs of and give explicit relations between them.

     

Periodic automorphisms of the free Lie algebra of rank 3

We prove that each automorphism of finite order of the free Lie algebra of rank 3 over an algebraically closed field is conjugate to a linear automorphism if the field characteristic fails to divide the automorphism order.     

The Mal'tsev-Nielsen equation in a free metabelian group of rank two

Translated from Matematicheskie Zametki, Vol. 46, No. 6, pp. 57–60, December, 1989.     

The degree of the special linear characters of a rank two free group

Given a word W in the free group on 2 letters F ₂, Horowitz showed that the special linear character of W is an integer polynomial in the 3 characters of the basic words of F ₂. Special linear characters are defined via the trace of their representations, and the polynomial character of an arbitrary W can be found by application of certain “trace relations”, which allow one to write the character of a complicated word as a sum of products of the characters of simpler words. Even in the n = 2 case, where the polynomial is uniquely defined, this procedure can be difficult and tedious. In this note, we use the structure of a free group word in F ₂ to compute the degree and the leading monomial of its SL(2,\mathbbC){SL(2,\mathbb{C})} -character without actually computing the full polynomial.     

Coordinates and automorphisms of polynomial and free associative algebras of rank three

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and algorithms that recognize z-tame automorphisms and z-tame coordinates.      

The algebraic entropy of the special linear character automorphisms of a free group on two generators

In this note, we establish a connection between the dynamical degree, or algebraic entropy of a certain class of polynomial automorphisms of , and the maximum topological entropy of the action when restricted to compact invariant subvarieties. Indeed, when there is no cancellation of leading terms in the successive iterates of the polynomial automorphism, the two quantities are equal. In general, however, the algebraic entropy overestimates the topological entropy. These polynomial automorphisms arise as extensions of mapping class actions of a punctured torus on the relative -character varieties of embedded in . It is known that the topological entropy of these mapping class actions is maximized on the relative character variety comprised of reducible characters (those whose boundary holonomy is ). Here we calculate the algebraic entropy of the induced polynomial automorphisms on the character varieties and show that it too solely depends on the topology of .

     

On soluble groups of module automorphisms of finite rank

Let R be a commutative ring, M an R-module and G a group of R-automorphisms of M, usually with some sort of rank restriction on G. We study the transfer of hypotheses between M/C _M (G) and [M,G] such as Noetherian or having finite composition length. In this we extend recent work of Dixon, Kurdachenko and Otal and of Kurdachenko, Subbotin and Chupordia. For example, suppose [M,G] is R-Noetherian. If G has finite rank, then M/C _M (G) also is R-Noetherian. Further, if [M,G] is R-Noetherian and if only certain abelian sections of G have finite rank, then G has finite rank and is soluble-by-finite. If M/C _M (G) is R-Noetherian and G has finite rank, then [M,G] need not be R-Noetherian.     

Cohomological dimension and symmetric automorphisms of a free group

The author gratefully acknowledges support from the Ruhr-Universit?t, Bochum and the Alexander von Humboldt-Foundation during the preparation of this paper.     

On free boundary problems in two dimensions

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ₀ of Γ, then Γ₀ permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.     

On the group of holomorphic automorphisms of ℂ n

Oblatum 21-IX-1991 & 30-IV-1992