Related papers: G.N. Chebotarev's "On the Klein-Hilbert Resolvent …

We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes…

This is an English translation of four remarkable articles, originally written in Russian, by Sergei Ivanovich Adian (1931--2020), supplemented by the translation of two closely related articles by Andrei Andreevich Markov Jr (1903--1979).…

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our…

This is an English translation of the thesis written by G. S. Makanin for the degree of Candidate of Physical and Mathematical Sciences (equivalent to a Ph.D.), originally submitted to the Steklov Mathematical Institute in 1966. The…

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple…

We solve Klein-Gordon equation (KGE) in the framework of the real Hilbert space approach to quaternionic quantum mechanics ($\mathbbm H$QM). The presented solution is the simplest ever obtained for quaternionic quantum theories, and the…

An overview of quantum computing and in particular the Hidden Subgroup Problem are presented from a mathematical viewpoint. Detailed proofs are supplied for many important results from the literature, and notation is unified, making it…

The paper contains a generalization of known properties of Chebyshev polynomials of the second kind in one variable to polynomials of $n$ variables based on the root lattices of compact simple Lie groups $G$ of any type and of rank $n$. The…

We construct a local deformation problem for residual Galois representations $\bar{\rho}$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles…

We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by…

A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a…

This paper gathers four lectures, based on a mini-course at IMA in 2014, whose aim was to discuss the structure of approximate subgroups of an arbitrary group, following the works of Hrushovski and of Green, Tao and the author. Along the…

Chaitin's incompleteness theorem states that sufficiently rich formal systems cannot prove lower bounds on Kolmogorov complexity. In this paper we extend this theorem by showing theories that prove the Kolmogorov complexity of a large (but…

To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear…

Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of…

We introduce a new notion of rank for unitary representations of semisimple groups over a local field of characteristic zero. The theory is based on Kirillov's method of orbits for nilpotent groups over local fields. When the semisimple…

This paper is concerned with the representation theory of finite groups. According to Robinson, the truth of certain variants of Alperin's weight conjecture on the $p$-blocks of a finite group would imply some arithmetical conditions on the…

All groups are 2-generator. For any prime-power q, Theorem 1 constructs a solvable matrix group over a quotient of a Laurent polynomial ring. This group is closely related to a group of exponent q as shown in Theorems 2 & 3 . Theorem 4 in…

The critical point between varieties A and B of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of A but of no member of B, if it exists. The study of critical points gives rise to a whole…

In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of…