Geometric methods in group theory have experienced a spectacular expansion since the work by Gromov and Thurston in the 1980s, with the development of hyperbolic and automatic groups, and the realization of the importance of spaces (including groups themselves) of nonpositive curvature. The research team coordinated by professor E. Ventura has been studying these problems for more than twenty years and its work is shaped in this field. The goals of the present project are dedicated to keep advancing in various aspects of the field. Furthermore, in some of them, special emphasis on computational and algorithmic aspects will be played. Among them we can highlight the following ones: - The study of certain algebraic properties of infinite groups, which are relevant from the geometric point of view: properties like residual finiteness, conjugacy class separability, amenability, or the study of the word metric for some groups, as well as subgroup distortion, provide very significant examples. Of special importance are limit groups, and hyperbolic and relatively hyperbolic groups, where geometric methods are crucial for their study. - The study of the lattice of subgroups of a free group sistematically using Stallings graphs: fixed subgroups under automorphisms or endomorphisms, retracts, inertia, algebraic extensions of subgroups, closures of subgroups under pro-C topologies, intersections of different types of subgroups, etc. We will also consider possible extensions of these concepts and goals into families which are close to free groups, like surface groups, virtually free groups, free products and amalgams of finite and cyclic groups, direct and semidirect products of free groups and abelian groups, some PC-groups, etc. - The study of algorithmic problems in several families of groups. Among them, we plan to find algorithms for the resolution of the three classical Dehn problems, especially the conjugacy problem, in different families or subfamilies of groups (free-by-cyclic, free-by-free, onerelator groups, hyperbolic groups, etc). We are also especially interested in the study of several algorithmic questions which are still open related to the lattice of subgroups of the free group (using the Stallings graphs mentioned above). These algorithmic methods have acquired a great deal of attention during the last few years due to their possible application to cryptographic methods based on groups. - The study of algebraic, geometric and algorithmic aspects of certain specific specially important groups like the braid group, Thomson's group and its variations, Houghton's groups, etc. Our research team has been considering several aspects of these groups in the last years, and it is our intention to keep going in this direction, studying some of the interesting open problems which still persist.
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia