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.     

The centralizer of a set of group automorphisms

Let A be a set of automorphisms of a finite group G. The structure of the near-ring C(A) of identity preserving functions which commute with every element of A is investigated.     

The group of automorphisms of a commutative algebra

Author partially supported by the DGICYT (PB90-0300-C02-02)     

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.     

Lower central series of a group of tree automorphisms

The lower central series of the Grigorchuk 3-generated 2-group is described; the indices of the factors of this series are calculated. The nilpotent length and solvability length of the factors by congruence subgroups are determined.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 225–237, August, 1996.     

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Homogeneous spaces generated by the group of automorphisms of a Lie group

The concept of a homogeneous space generated by the group of automorphisms of a given Lie group generalizes the well-known concept of a-space. We study the properties of such spaces by analogy with-spaces.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 15, pp. 165–185, 1983.     

Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.     

Lie algebras admitting a metacyclic frobenius group of automorphisms

Suppose that a Lie algebra L admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H such that the characteristic of the ground field does not divide |H|. It is proved that if the subalgebra C _L (F) of fixed points of the kernel has finite dimension m and the subalgebra C _L (H) of fixed points of the complement is nilpotent of class c, then L has a nilpotent subalgebra of finite codimension bounded in terms of m, c, |H|, and |F| whose nilpotency class is bounded in terms of only |H| and c. Examples show that the condition of F being cyclic is essential.     

Some families of polynomial automorphisms

We give a family of polynomial automorphisms of the complex affine plane whose generic length is 3 and degenerating in an automorphism of length 1 with surprisingly high degree.     

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.     

Some automorphisms of Laguerre planes

Results in Mathematics -     

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shiftV _β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV _β and certainα-invariant probability measures onX. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.