We study automorphisms φ of the free associative algebra K〈x, y, z〉 over a field K such that φ(x), φ(y) are linear with respect to x, y and φ(z) = z. We prove that some of these automorphisms are wild in the class of all automorphisms fixing z, including the well known automorphism discovered by Anick, and show how to recognize the wild ones. This class of automorphisms induces tame automorphis...

Let K be a field of characteristic zero. We characterize coordinates and tame coordinates in K[z][x,y], i.e. the images of x respectively under all automorphisms and under the tame automorphisms of K[z][x,y]. We also construct a new large class of wild automorphisms of K[z][x,y] which maps x to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, ...

We investigate the question of when an ergodic automorphism T is conjugate to its composition square T 2, i.e., when does there exist an automorphism S with the property that ST = T 2S. This is a non–generic property of automorphisms which seems to be quite exceptional. The situation for ergodic automorphisms having discrete spectrum and automorphisms having the weak closure property is investi...

We classify isotopy classes of automorphisms (self-homeomorphisms) of 3manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as a suitable composition of two automorphisms, each of which fits into our clas...

Let W be a variety of groups defined by a set W of laws and G be a finite p-group in W. The automorphism α of a group G is said to bea marginal automorphism (with respect to W), if for all x ∈ G, x−1α(x) ∈ W∗(G), where W∗(G) is the marginal subgroup of G. Let M,N be two normalsubgroups of G. By AutM(G), we mean the subgroup of Aut(G) consistingof all automorphisms which centralize G/M. AutN(G) ...

Let G be a connected reductive group. We find a necessary and sufficient condition for a quasiaffine homogeneous space of G/H to be embeddable into an irreducible G-module. If H is reductive we also find a necessary and sufficient condition for a closed embedding of G/H into an irreducible module to exist. These conditions are stated in terms of the group of central automorphisms of G/H .

Kleinewillinghöfer classified Laguerre planes with respect to central automorphisms. Polster and Steinke investigated flat Laguerre planes and their so-called Kleinewillinghöfer types, that is, the Kleinewillinghöfer types with respect to the full automorphism group. For some of these types the existence question remained open. We provide strong necessary existence conditions for flat Laguerre ...

In 2004, Shestakov and Umirbaev proved that the Nagata automorphism of polynomial algebra in three variables is wild. We fix a [Formula: see text]-grading on this consider graded-wild automorphisms, i.e. such automorphisms cannot be decomposed onto elementary respecting grading. describe all gradings allowing automorphisms.

We determine which simple algebraic groups of type D4 over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in terms of a representation of the group by automorphisms of a trialitarian algebra: outer automorphisms of order 3 exist if and only if the algebra is the endomorphis...

Several results on the action of graph automorphisms on ends and fibers are generalized for the case of metric ends. This includes results on the action of the automorphisms on the end space, directions of automorphisms, double rays which are invariant under a power of an automorphism and metrically almost transitive automorphism groups. It is proved that the bounded automorphisms of a metrical...