Polynomial structures on polycyclic groups

Polynomial Structures for Nilpotent Groups

If a polycyclic-by-finite rank-K group Γ admits a faithful affine representation making it acting on RK properly discontinuously and with compact quotient, we say that Γ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups Γ. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent group...

On Ordered Polycyclic Groups

It has been asserted that any (full) order on a torsion-free, finitely generated, nilpotent group is defined by some F-basis of G and that the group of o-automorphisms of such a group is itself a group of the same kind: Examples provided herein demonstrate that both of these assertions are false; however, it is proved that the group of o-automorphisms of an ordered, polycyclic group is nilpoten...

Aperiodic Subshifts on Polycyclic Groups

We prove that every polycyclic group of nonlinear growth admits a strongly aperiodic SFT and has an undecidable domino problem. This answers a question of [5] and generalizes the result of [2]. Subshifts of finite type (SFT for short) in a group G are colorings of the elements of G subject to local constraints. They have been studied extensively for the free abelian group Z [14] and on the free...

A Survey on Omega Polynomial of Some Nano Structures

Elementary Abelian Hopf Galois Structures and Polynomial Formal Groups

We find a relationship between regular embeddings of G, an elementary abelian p-group of order p, into unipotent upper triangular matrices with entries in Fp and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices. We classify the latter when n = 3 up to linear isomorphism, and use that classification to determine the number of Hopf Galoi...