Units in orders of finite dimensional algebras

• Grant number: 11/11315-7
• Support Opportunities: Regular Research Grants
• Start date: September 01, 2011
• End date: August 31, 2013
• 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
• Associated researchers: Antônio de Andrade e Silva ; Edson Ryoji Okamoto Iwaki ; Eric Jespers ; Orlando Stanley Juriaans ; Wolfgang Kimmerle

Abstract

In the article \cite{jpsf}, we classified the finite groups $G$ and rational quadratic extensions $K=\Q(\sqrt{-d})$, such that the group of units of the group ring $\oo_k G$ be a hyperbolic group. We have that the group of units of the integral algebraic integers of $(\frac{-1,-1}{K})$, say $(\frac{-1,-1}{\oo_K})$, is a co-compact discret group, when $d \equiv 7 \pmod 8$ and $K$ is imaginary. Defining an equivalence relation in this group, using a well known embedding ({\it a ray}), we show that there exists a unique equivalence class. Then we say that the number of Ends of this group is $1$. In the articles \cite{ijsf,ijsii}, we classified the finite dimensional associative $\Q$-algebras $A$ with a property that for any $\Z$-order $\Lambda \in A$, the unit group $\U(\Lambda)$ has no subgroup isomorphic to a free Abelian group of rank $2$, which was coined as {\it hyperbolic property}. Recently this result was extended to finite dimensional alternative $\Q$-algebras $\A$, see \cite{jposf}. As a result, we show that the radical of $\A$ lies in its associator. From these results, some questions remain not answered. For example, if $G$ is a finitely generated group, although the number of ends of $G$ is $0,1,2$ or $\infty$, we want to determine the number of ends of certain classes of groups, as we did it to the unit group of $(\frac{-1,-1}{\oo_K})$, where $d \equiv 7 \pmod 8$. Also, we want to study the finite dimensional algebras $A$ with the hyperbolic property and which the unit group of the orders of the algebra is a hyperbolic group. Other problem we propose to study is the classification of finite groups $G$ which the unit group $\U(\Z G)$ has no subgroups isomorphic to free Abelian groups of rank $3$. Close to this question, also it seems important to determine the structure of finite dimensional alternative $\Q$-algebras whose radical lies in the associator of the algebra. We are also completing one of the problems presented in the project $2008/57930-1$ on the generating units of the unit group of the ring of integers $ \Gamma_d = \h (\Z (\frac{1 + \sqrt {-d}} {2} )) $ of the quaternion algebra over $ \Q (\sqrt{-d}) $, where $ d \equiv 7 \pmod $ 8. Problems of this type have been subject of research for years and only recently, when $ d = 7$ a finite set of generators were firstly presented in the article \cite{cjlr}. Using a method related to construction of a domain of Ford, in our work, we present a formulation that allows to determine generators for the group $\U(\Gamma_d)$. We show examples for the cases $ d = 15$ and $d=23$. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

JURIAANS, S. O.; SILVA, A. DE A. E.; SOUZA FILHO, A. C.. Traces of Torsion Units. COMMUNICATIONS IN ALGEBRA, v. 43, n. 9, p. 3925-3933, 2015. (11/11315-7)

JESPERS, E.; JURIAANS, S. O.; KIEFER, A.; DE A E SILVA, A.; SOUZA FILHO, A. C.. FROM THE POINCARE THEOREM TO GENERATORS OF THE UNIT GROUP OF INTEGRAL GROUP RINGS OF FINITE GROUPS. Mathematics of Computation, v. 84, n. 293, p. 1489-1520, MAY 2015. (11/11315-7)