Valuation theory of group rings and homology of soluble groups

• Grant number: 16/13937-9
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): November 01, 2016
• Effective date (End): July 27, 2020
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Daniel Levcovitz
• Grantee: Fatemeh Yeganeh Mokari
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated scholarship(s): 18/03561-7 - Homology of linear groups over Dedekind domains, BE.EP.PD

Abstract

Groups and rings are basic but fundamental algebraic structures in mathematics and they appear almost in all subjects of Mathematics. Groups have inseparable connection with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object. Rings are natural generalization of the set of integer numbers. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions. The main purpose of this project is to use the tools available in rings theory, such as module theory and valuation theory of group rings to study certain deep problems in homological aspect of group theory. This is very powerful method to study groups. Investigation in this direction has led to the discovery of many interesting connections and to the creation of new powerful methods of investigation and has come to involve a large collection of mathematical tools and view points. An important class of groups is the class of soluble groups. Finitely generated soluble groups occurring in applications are often nilpotent-by-abelian-by-finite. For example S-arithmetic soluble groups are nilpotent-by-abelian-by-finite. One of the main plans of this project is to study homological aspects of nilpotent-by-abelian groups and the most important invariant for the study of these groups is the geometric invariant of Bieri-Strebel. This is a geometric invariant and using it one can define n-tameness of modules over group rings. The n-tameness of an action of a finitely generated abelian group Q over a Q-module has a deep connection with valuation theory and module theory of the group ring of Q. Although there has been much important progress in this direction, still there are many unanswered questions. This part of our project involves a lot of commutative algebra and we hope that our knowledge in this subject, in particular valuation theory, would help us with some deeper understanding of tameness. Furthermore, the above subjects are connected to the Betti numbers of groups, which are very important invariant of groups. The virtual Betti number of a finitely generated group studies the growth of the Betti numbers of a group as one follows passage to subgroups of finite index. Finiteness of virtual Betti numbers of a group is a very strange property. It is interesting to know that what phenomena lie behind this property of some groups. In this project we plan to study virtual Betti numbers of solvable groups. In many respects nilpotent groups behave very similar to abelian groups. We plan to see how far this can go when one studies homology of these groups. Moreover, there are many properties of homology of abelian groups that one expects as well to be true for homology of nilpotent groups, but surprisingly this is not the case. We plan to investigate such problem in order to get better understanding of the homology of nilpotent and hopefully of solvable groups. Along the line, we plan to investigate homology of solvable groups. For example, what is the exact structure of the integral homology of metabelian groups. This is not known even for the second homology of such groups. In this project three major subjects of Mathematics, i.e. groups theory, ring theory and algebraic topology come together. With the approaches and strategies that is explained in our research plan we hope to answer some fundamental conjectures or make a substantial contribution toward their solution. (AU)