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Geometric Methods in Group Theory

Type of activity
Competitive project
Funding entity
Funding entity code
18.029,00 €
Start date
End date
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 fifteen
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
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 products of free
groups and abelian 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 above mentioned Stallings graphs). 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
- The study of algebraic, geometric and algorithmic aspects of certain specific specially important groups like the Braid group, Thomson's
group, Hoghton's group, 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 presist.
Adm. Estat
Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016
Call year
Funcding program
Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia
Funding subprogram
Subprograma Estatal de Generación de Conocimiento
Funding call
Excelencia: Proyectos I+D
Grant institution
Gobierno De España. Ministerio De Economía Y Competitividad, Mineco


  • Ventura Capell, Enric  (scientific coordinator)
  • Lopez Platon, Eric  (researcher)
  • Burillo Puig, Jose  (researcher)
  • Delgado Rodriguez, Jorge  (researcher)
  • Cleary, Sean  (researcher)
  • Wu, Jianchun  (researcher)
  • Antolín, Yago  (researcher)
  • Martino, Armando  (researcher)