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Let D be a division ring with center F and N a subnormal subgroup of the multiplicative group D* of D. Assume that N contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in N. We show that if either N is algebraic over F or F is uncountable, then N contains a non-abelian free subgroup.

Let k be a field of characteristic different from 2 and let G be a nonabelian residually torsion-free nilpotent group. It is known that G is an orderable group. Let k(G) denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of G over k. If ∗ is an involution on G, then it extends to a unique k-involution on k(G). We show that k(G) contains pairs of symmetric elements with respect to ∗ which generate a free group inside the multiplicative group of k(G). Free unitary pairs also exist if G is torsion-free nilpotent. Finally, we consider the general case of a division ring D, with a k-involution ∗, containing a normal subgroup N in its multiplicative group, such that G⊆N , with G a nilpotent-by-finite torsion-free subgroup that is not abelian-by-finite, satisfying G∗ = G and N∗ = N. We prove that N contains a free symmetric pair.

A It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear algebraic group of relative rank one over an arbitrary field We also obtain explicit formulas for the root groups, the

Let R be a ring and let $n\\ge 2$ . We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$ , for $x,y\\in M_n(R)$ . An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\\in M_n(R)$ , then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$ .

Let D be a division ring and N be a subnormal subgroup of the multiplicative group $D^*$ . We show that if N contains a nonabelian solvable subgroup, then N contains a nonabelian free subgroup.

We study the discrete dynamics of standard (or left) polynomials $f(x)$ over division rings D. We define their fixed points to be the points $\\lambda \\in D$ for which $f^{\\circ n}(\\lambda )=\\lambda $ for any $n \\in \\mathbb {N}$ , where $f^{\\circ n}(x)$ is defined recursively by $f^{\\circ n}(x)=f(f^{\\circ (n-1)}(x))$ and $f^{\\circ 1}(x)=f(x)$ . Periodic points are similarly defined. We prove that $\\lambda $ is a fixed point of $f(x)$ if and only if $f(\\lambda )=\\lambda $ , which enables the use of known results from the theory of polynomial equations, to conclude that any polynomial of degree $m \\geq 2$ has at most m conjugacy classes of fixed points. We also show that in general, periodic points do not behave as in the commutative case. We provide a sufficient condition for periodic points to behave as expected.

Hua's fundamental theorem of geometry of matrices describes the general form of bijective maps on the space of all m\\times n matrices over a division ring \\mathbb{D} which preserve adjacency in both directions. Motivated by several applications the author studies a long standing open problem of possible improvements. There are three natural questions. Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only and still get the same conclusion? Can we relax the bijectivity assumption? Can we obtain an analogous result for maps acting between the spaces of rectangular matrices of different sizes? A division ring is said to be EAS if it is not isomorphic to any proper subring. For matrices over EAS division rings the author solves all three problems simultaneously, thus obtaining the optimal version of Hua's theorem. In the case of general division rings he gets such an optimal result only for square matrices and gives examples showing that it cannot be extended to the non-square case.

We show that a virtually residually finite rationally solvable (RFRS) group $G$ of type $\\mathtt {FP}_n(\\mathbb {Q})$ virtually algebraically fibres with kernel of type $\\mathtt {FP}_n(\\mathbb {Q})$ if and only if the first $n$ $\\ell ^2$-Betti numbers of $G$ vanish, that is, $b_p^{(2)}(G) = 0$ for $0 \\leqslant p \\leqslant n$. This confirms a conjecture of Kielak. We also offer a variant of this result over other fields, in particular in positive characteristic. As an application of the main result, we show that amenable virtually RFRS groups of type $\\mathtt {FP}(\\mathbb {Q})$ are virtually Abelian. It then follows that if $G$ is a virtually RFRS group of type $\\mathtt {FP}(\\mathbb {Q})$ such that $\\mathbb {Z} G$ is Noetherian, then $G$ is virtually Abelian. This confirms a conjecture of Baer for the class of virtually RFRS groups of type $\\mathtt {FP}(\\mathbb {Q})$, which includes (for instance) the class of virtually compact special groups.

We extend a generalization of Artin’s Theorem to alternative division rings. We characterize maps between alternative division rings taking products equal to one fixed element to products equal to another fixed element and study when these maps are either automorphisms or anti-automorphisms. In particular, we completely describe these maps in the case of Jordan homomorphisms.