Inspired by the Basilica group $\mathcal B$, we describe a general construction which allows us to associate any of automorphisms $G \leq \operatorname{Aut}(T)$ rooted tree $T$ family groups $\operatorname{Bas}s(G)$, $s \in \mathbb{N}+$. For dyadic odometer O\_2$, one has B = \operatorname{Bas}\_2(\mathcal O\_2)$. We study properties acting on trees are preserved under this operation. Introduci...

We give a necessary and sufficient condition for k-step nilmanifolds associated with graphs (k ≥ 3) to admit Anosov automorphisms. We also prove nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds. 2000 Mathematics Subject Classification. Primary 37D20; Secondary 22E25

The notion of Cayley color graphs of groups is generalized to inverse semigroups and groupoids. The set of partial automorphisms of the Cayley color graph of an inverse semigroup or a groupoid is isomorphic to the original inverse semigroup or groupoid. The groupoid of color permuting partial automorphisms of the Cayley color graph of a transitive groupoid is isomorphic to the original groupoid.

In the paper we study the endomorphim semigroup of a general quantum polynomial ring, its finite groups of automorphisms and homological properties of this ring as a module over the skew group ring of a finite group of automorphisms. Moreover properties of the division ring of fractions are considered.

We study whether the basin of attraction of a sequence of automorphisms of C is biholomorphic to C. In particular we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to C if the maps are repeated often enough. We also construct Fatou-Bieberbach domains whose boundaries are 4-dimensional.

We classify certain automorphisms of an even unimodular lattice L with fixed irreducible and unramified characteristic polynomial. The question which automorphisms are conjugate to their inverses in the orthogonal group of L is investigated. 2000 Math. Subj. Class. 11H56, 11R06, 20F55.

Given a graphG = (V,E) with a setW ⊆ V of vertices, we enumerate colorings to W such that for every two enumerated colorings c and c′ the corresponding colored graphs (G, c) and (G, c′) are not isomorphic. This problem has an important application in the study of isomers of chemical graphs such as generation of benzen isomers from a tree-like chemical graph structure. The number of such colorin...

Imposing a group structure on the set of random orders, the paper reformulates and characterizes the random order value and more generally semi-value operators in a uni ed framework that encompasses games with nite and continuum of players and allows symmetry of the operators to be with respect to a subgroup of automorphisms. A set of orderings of players equipped with a group structure induced...

The main objects of this paper are the graded automorphisms of w x polytopal semigroup rings, i.e., semigroup rings k S where k is a field P Ž and S is the semigroup associated with a lattice polytope P Bruns, P w x. w x Gubeladze, and Trung BGT . The generators of k S correspond bijecP tively to the lattice points in P, and their relations are the binomials representing the affine dependencies...

The McKay–Navarro conjecture is a refinement of the McKay that additionally takes action some Galois automorphisms into account. We verify inductive condition in defining characteristic for finite groups Lie type with exceptional graph automorphisms, Suzuki and Ree groups, Bn(2) (n?2), non-generic Schur multiplier. This completes verification their characteristic.