Units of group rings, semigroup rings and orders in finite dimensional algebras
Groups and noncommutative algebra: interactions and applications
Grant number: | 14/06325-1 |
Support Opportunities: | Regular Research Grants |
Duration: | October 01, 2014 - September 30, 2016 |
Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Algebra |
Principal Investigator: | Antonio Calixto de Souza Filho |
Grantee: | Antonio Calixto de Souza Filho |
Host Institution: | Escola de Artes, Ciências e Humanidades (EACH). Universidade de São Paulo (USP). São Paulo , SP, Brazil |
Abstract
In \cite{grmv} throughout the concept of hyperbolic spaces, Gromov defined the Hyperbolic groups. For a given finitely generated group it is possible to construct a metric which associated to the Cayley graph of the group define a metric space. A group is hyperbolic if its Cayley graph is a hyperbolic metric space. In \cite{jpp} are studied the finite groups whose its integral group ring are hyperbolic. In this caracterization, the Flat Plane Theorem plays an essential role. This strategy shows that it is importante to know the Abelian free rank of a group, i. e., what is the biggest free rank of an abelian group embeddind in that group. This approach has raised many questions as well as it has contributed to the researche of the hyperbolicity of the unit group of $\Z$-orders of finite dimensinal algebras, as instance group algebras over infinite fields. In this way, is was classified the ring of algebraic integers of quadratic algebras over the rational numbers, and also the finite groups which the unit group of the group ring be a hyperbolic group, \cite{jpsf}. We also classify the finite dimensional algebras whose unit group of any $\Z$-order of the algebra be a hyperbolic group for semisimple or non trivial radical algebras, \cite{ijsf}. These results were applied on algebraic structures such as loop and semisimple algebras with analogous approach, but we needed to define the notion of {\it hyperbolic property} which is related to the Flat Plane Theorem. Thus, also it was characterized the finite semigroups which its semigroup algebra has the hyperbolic property, \cite{ijsf,ijsii}, as well as the loop whose unit loop of the integral loop ring has the same property, \cite{jsfAA}. In this same research program, from a result of Gromov on hyperbolic groups with free groups, in \cite{jsf} it was constructed pair of units (and not even their existence) which generate a free group in $\Z$-orders of quaternions algebra over rational quadratic extensions. Such result is realizable because, according with the technics in \cite{jpsf} for the class of these algebras we construct two new classes of units: Pell's, and Gauss', units. The feature of these units jointly with the results of \cite{cjlr} started a new investigation on the possibility to extend the results of this paper, among them, to consider another algebraic integral ring of the quadratic extension studied there. As a result, we present generating sets of the unit group of that $\Z$-order when $d \in \{15,23\}$, among others progress, \cite{geoag,geo}. (AU)
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