Condensed groups in product varieties

A finitely generated group 𝐺 is said to be condensed if its isomorphism class in the space of finitely generated marked groups has no isolated points. We prove that every product variety U ⁢ V \mathcal{UV} , where 𝒰 (respectively, 𝒱) is a non-abelian (respectively, a non-locally finite) variety, contains a condensed group. In particular, there exist condensed groups of finite exponent. As an application, we obtain some results on the structure of the isomorphism and elementary equivalence relations on the set of finitely generated groups in U ⁢ V \mathcal{UV} .

IDENTITIES IN BRANDT SEMIGROUPS, REVISITED

We present a new proof for the main claim made in the author's paper "On the identity bases of Brandt semigroups" (Ural. Gos. Univ. Mat. Zap., 14, no.1 (1985), 38–42); this claim provides an identity basis for an arbitrary Brandt semigroup over a group of finite exponent. We also show how to fill a gap in the original proof of the claim in loc. cit.

Kp-series and varieties generated by wreath products of p-groups

Let [Formula: see text] be a nilpotent [Formula: see text]-group of finite exponent and [Formula: see text] be an abelian [Formula: see text]-group of finite exponent for a given prime number [Formula: see text]. Then the wreath product [Formula: see text] generates the variety [Formula: see text] if and only if the group [Formula: see text] contains a subgroup isomorphic to the direct product [Formula: see text] of countably many copies of the cycle [Formula: see text] of order [Formula: see text]. The obtained theorem continues our previous study of cases when [Formula: see text] holds for some other classes of groups [Formula: see text] and [Formula: see text] (abelian groups, finite groups, etc.).

Transformations preserving the Lyapunov exponents

We give a complete characterization of the existence of Lyapunov coordinate changes bringing an invertible sequence of matrices to one in block form. In other words, we give a criterion for the block-trivialization of a nonautonomous dynamics with discrete time while preserving the asymptotic properties of the dynamics. We provide two nontrivial applications of this criterion: we show that any Lyapunov regular sequence of invertible matrices can be transformed by a Lyapunov coordinate change into a constant diagonal sequence; and we show that the family of all coordinate changes preserving simultaneously the Lyapunov exponents of all sequences of invertible matrices (with finite exponent) coincides with the family of Lyapunov coordinate changes.

State-closed groups with bounded conjugacy classes

Let [Formula: see text] be the group of automorphisms of the one-rooted [Formula: see text]-ary tree and [Formula: see text] be a transitive state-closed subgroup of [Formula: see text] with bounded finite conjugacy classes. We prove that the torsion subgroup Tor[Formula: see text] has finite exponent and determine an upper bound for the exponent. In case [Formula: see text] is a prime number, we prove that [Formula: see text] is either a torsion group or a torsion-free abelian group.

On the Northcott property and other properties related to polynomial mappings

We prove that if K/ℚ is a Galois extension of finite exponent and K(d) is the compositum of all extensions of K of degree at most d, then K(d) has the Bogomolov property and the maximal abelian subextension of K(d)/ℚ has the Northcott property.Moreover, we prove that given any sequence of finite solvable groups {Gm}m there exists a sequence of Galois extensions {Km}m with Gal(Km/ℚ)=Gm such that the compositum of the fields Km has the Northcott property. In particular we provide examples of fields with the Northcott property with uniformly bounded local degrees but not contained in ℚ(d).We also discuss some problems related to properties introduced by Liardet and Narkiewicz to study polynomial mappings. Using results on the Northcott property and a result by Dvornicich and Zannier we easily deduce answers to some open problems proposed by Narkiewicz.