LCCRM · Equations in free groups
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2005-01-01 → 2006-12-31
- EU contribution
- €133,046
- Participants
- 1
- Scheme
- EIF
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Results in brief
Final Activity Report Summary - LCCRM (Equations in free groups)
The topic of the project was the study of equations in free groups from a variety of perspectives: combinatorial, algorithmic, algebraic and geometric. Solving equations in groups and monoids has been explored by many mathematicians and computer scientists. This problem can be tackled algorithmically and expressed via algebraic geometry over groups. Also, solving equations is a key ingredient in the study of the elementary and universal theories of a group, and shows up in theoretical computer science in the context of unification theory. Within this project the fellow has worked on many interconnected aspects of free groups. In the area of decision problems and their complexity we mention the paper 'Polynomial-time complexity for instances of the endomorphism problem in free groups', which has appeared in the International Journal of Algebra and Computation. This paper establishes, among other things, that there is a polynomial-time algorithm for deciding the solvability, in free groups, of two-variable equations in which all the variables occur on one side of the equality and all the constants on the other side. Furthermore, regarding equations and their solutions, the fellow together with Sasa Radomirovic, has produced the paper 'Restricted walks in regular trees, which has appeared in the Electronic Journal of Combinatorics. The analysis of equations and their solutions also played a key role in the study of Galois theory of free groups. The fellow has produced the paper 'Two examples in the Galois theory of free groups' with Prof. Warren Dicks, the scientist in charge, which has appeared in the Journal of Algebra. A related direction of work has been expanding some of the known results for free groups to much wider classes of groups, via group actions on trees and generalisations of trees. In this direction, the fellow, together with N.Brady, A. Martino and S. O Rourke, has written the paper 'The equation x^py^q=s^r in groups that act freely on Lambda-trees', which will appear in the Transactions of the American Mathematical Society. The fellow has participated in more than a dozen conferences during the time of her fellowship, and presented her work at most of these events. Also, the fellow has been invited to give talks in numerous seminars in Spain, United States, France, Germany and Switzerland. The fellow will continue to interact with the group theorists in Barcelona: Enric Ventura, Jose Burillo and Armando Martino, with whom she has started joint projects during the stay at the Centre de Recerca Matematica, stay facilitated by the Marie Curie Fellowship.
Data: CORDIS, © European Union
Project objective
In this project we wish to explore theoretical, as well as practical, aspects of solving equations in free groups. We will first consider a particular type of equation that occurs while solving the endomorphism problem in free groups. We say that the endom orphism problem is solvable for an element W in a group G if it can be decided effectively whether or not, given U in G, there is an endomorphism of G sending Wto U. We aim to display a polynomial-time algorithm that solves the endomorphism problem in free groups and also analyse the endomorphism problem from the perspective of generic-time complexity as developed by Schupp, Kapovich, Myasnikov and Shpilrain.Another goal of this project is the classification of equations in free groups in the following sense. We call a group G separable with respect to a class of equations if, for every equation in the class with no solution in G, there is a finite quotient of G in which the equation has no solution. We want to determine what classes of equations the free groups are separable with respect to.
Original text from CORDIS.
Participants
- CENTRE DE RECERCA MATEMATICA · BELLATERRACoordinatorCity levelSpain
Links
Data: CORDIS, © European Union