Mathematics

We establish a sufficient condition for a finitely generated pro- p group to be accessible in terms of finite generation of the module of ends.

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 original language is Russian. The named supervisors are A. A. Markov and S. I. Adian.

A loop X is said to satisfy Moufang's theorem if for every x,y,z?�X such that x(yz)=(xy)z the subloop generated by x , y , z is a group. We prove that the variety V of Steiner loops satisfying the identity (xz)(((xy)z)(yz))=((xz)((xy)z))(yz) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah.

A direct consequence of Gromov's theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.

We obtain a simple family of solutions to the set-theoretic Yang-Baxter equation, one which depends only on considering special endomorphisms of a finite group. We show how such an endomorphism gives rise to two non-degenerate solutions to the Yang-Baxter equation, solutions which are inverse to each other. We give concrete examples using dihedral, alternating, symmetric, and metacyclic groups.

Let G be a finite nonabelian group, and let ψ:G→G be a homomorphism with abelian image. We show how ψ gives rise to two Hopf-Galois structures on a Galois extension L/K with Galois group (isomorphic to) G ; one of these structures generalizes the construction given by a ``fixed point free abelian endomorphism'' introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

Let G be a finite nonabelian group. We show how an endomorphism of G with abelian image gives rise to a family of binary operations { ??n :n??Z ?? } on G such that (G, ??m , ??n ) is a skew left brace for all m,n?? . A brace block gives rise to a number of non-degenerate set-theoretic solutions to the Yang-Baxter equation. We give examples showing that the number of solutions obtained can be arbitrarily large.

We classify abelian subgroups of two-dimensional Artin groups.

It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.