A generic distal tower of arbitrary countable height over an arbitrary infinite ergodic system

<p style='text-indent:20px;'>We show the existence, over an arbitrary infinite ergodic <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-dynamical system, of a generic ergodic relatively distal extension of arbitrary countable rank and arbitrary infinite compact extending groups (or more generally, infinite quotients of compact groups) in its canonical distal tower.</p>

Non-∀-homogeneity in free groups

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not [Formula: see text]-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve [Formula: see text]-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.

Cellular covers of ℵ1-free abelian groups

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

Self-Similar Automorphisms of a Free Group of Countable Rank

We investigate self-similar automorphisms of a free group $F$ of infinite countable rank, that is automorphisms for which their actions on $F$ and $F^{\prime}$ are similar. We show properties, examples and counterexamples of self-similar automorphisms and study the subgroup generated by self-similar automorphisms.

GENERALIZED PRESENTATIONS OF INFINITE GROUPS, IN PARTICULAR OF Aut(Fω)

We develop a theory of generalized presentations of groups and give a generalized presentation of the automorphism group of the free group of countable rank, Aut (Fω).

THE FREE ALTERNATIVE SUPERALGEBRA ON ONE ODD GENERATOR

A base of the free alternative superalgebra on one odd generator is constructed. As a corollary, a base of the alternative Grassmann algebra is given. We also find a new element of degree 5 from the radical of the free alternative algebra of countable rank.