arXiv subjects

Jairo Z. Goncalves

Publications and source records attributed to Jairo Z. Goncalves.

5 recordsLinked to original sources

arXiv2018

Free symmetric and unitary pairs in the field of fractions of torsion-free nilpotent group algebras

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 $\ast$ 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 $\ast$ 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 $\ast$, containing a normal subgroup $N$ in its multiplicative group, such that $G \subseteq N$, 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.

math.RA↗

arXiv2015

Free algebras and free groups in Ore extensions and free group algebras in division rings

Let $K$ be a field of characteristic zero, let $σ$ be an automorphism of $K$ and let $δ$ be a $σ$-derivation of $K$. We show that the division ring $D=K(x;σ,δ)$ either has the property that every finitely generated subring satisfies a polynomial identity or $D$ contains a free algebra on two generators over its center. In the case when $K$ is finitely generated over $k$ we then see that for $σ$ a $k$-algebra automorphism of $K$ and $δ$ a $k$-linear derivation of $K$, $K(x;σ)$ having a free subalgebra on two generators is equivalent to $σ$ having infinite order, and $K(x;δ)$ having a free subalgebra is equivalent to $δ$ being nonzero. As an application, we show that if $D$ is a division ring with center $k$ of characteristic zero and $D^*$ contains a solvable subgroup that is not locally abelian-by-finite, then $D$ contains a free $k$-algebra on two generators. Moreover, if we assume that $k$ is uncountable, without any restrictions on the characteristic of $k$, then $D$ contains the $k$-group algebra of the free group of rank two.

math.RA↗

arXiv2014

Free symmetric and unitary pairs in division rings infinite-dimensional over their centers

Let $D$ be a division ring infinite-dimensional over its center $k$ with multiplicative group $D^{\times}$. We show that if $D$ belongs to certain families, there exist free symmetric and unitary pairs in $D^{\times}$ with respect to a $k$-involution on $D$.

math.RA↗

arXiv2012

Free symmetric group algebras in division rings generated by poly-orderable groups

We show that the canonical involution on a nonabelian poly-orderable group G extends to the Hughes-free division ring of fractions D of the group algebra k[G] of G over a field k and that, with respect to this involution, D contains a pair of symmetric elements freely generating a free group subalgebra of D over k.

math.RA↗

arXiv2007

Bicyclic units, Bass cyclic units and free groups

Let $G$ be a finite group and $\Z G$ its integral group ring. We show that if $α$ is a non-trivial bicyclic unit of $\Z G$, then there are bicyclic units $β$ and $γ$ of different types, such that $\GEN{α,β}$ and $\GEN{α,γ}$ are non-abelian free groups. In case that $G$ is non-abelian of order coprime with 6, then we prove the existence of a bicyclic unit $u$ and a Bass cyclic unit $v$ in $\Z G$, such that for every positive integer $m$ big enough, $\GEN{u^m,v}$ is a free non-abelian group.

math.RA↗