Constructing uncountably many groups with the same profinite completion

Two constructions are described: one gives soluble groups of derived length 4, the other uses groups acting on a rooted tree.

Subgroups of elliptic elements of the Cremona group

The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.

A classification of the abelian minimal closed normal subgroups of locally compact second-countable groups

We classify the locally compact second-countable (l.c.s.c.) groups 𝐴 that are abelian and topologically characteristically simple. All such groups 𝐴 occur as the monolith of some soluble l.c.s.c. group 𝐺 of derived length at most 3; with known exceptions (specifically, when 𝐴 is \mathbb{Q}^{n} or its dual for some n\in\mathbb{N}), we can take 𝐺 to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.

A NOTE ON DERIVED LENGTH AND CHARACTER DEGREES

Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$ . We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang

Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

Groups with restricted non-permutable subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be permutable if [Formula: see text] for every subgroup [Formula: see text] of [Formula: see text] and the group [Formula: see text] is called metaquasihamiltonian if all subgroups of [Formula: see text] are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group [Formula: see text] is soluble with derived length at most [Formula: see text] and contains a finite normal subgroup [Formula: see text] such that all subgroups of the factor [Formula: see text] are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian.