Profinite topology on non-positively curved groups

Geometric Group Theory is a vast area of Mathematics that combines ideas from Algebra, Analysis, Geometry and Topology and makes important contributions to all of these subjects. This area has been rapidly developing during the last 20 years, and has become popular among mathematicians all around the world. One of the principal themes of Geometric Group Theory is the study of non-positively curved groups. A group G is non-positively curved if it acts by transformations (in a sufficiently good manner) on a space X, whose geometry is similar to the geometry of a Euclidean or Hyperbolic space. In the presence of such an action, the properties of X give a lot of information about the structure of G and vice-versa.One of the most natural ways to study an infinite discrete group G is to look at its finite quotients. However, in general much of the information about G cannot be recovered this way; e.g., there exist infinite groups which have no non-trivial finite quotients at all. This is the classical reason for introducing the following two properties of G. The group G is said to be residually finite if for any two distinct elements x,y in G, there is a finite quotient-group Q of G such that the images of x and y are distinct in Q. And G is called conjugacy separable if for any two non-conjugate elements x,y in G there is a homomorphisms from G to finite group Q which maps x and y to non-conjugate elements of Q. Residual finiteness and conjugacy separability are natural combinatorial analogues of solvability of the word and conjugacy problems respectively. Indeed, a classical theorem of Mal'cev asserts that a finitely presented residually finite [conjugacy separable] group has solvable word problem [conjugacy problem]. Many groups are easily shown to be residually finite; on the other hand, proving that a group is conjugacy separable is a much more difficult task. Until recently, conjugacy separability was known for only a few families of groups.The proposed project aims to prove conjugacy separability for large classes of non-positively curved groups and establish residual finiteness for their automorphism groups. Its outcome will improve our understanding of the connection between geometric and algebraic properties of non-positively curved groups, and will shed some light on outstanding open problems in Geometric Group Theory.In a recent paper the PI proved that right angled Artin groups, forming an important subclass of non-positively curved groups, and all of their finite index subgroups are conjugacy separable. The significance of this algebraic theorem becomes clear after combining it with geometric results of Haglund and Wise, which provides an abundance of new examples of conjugacy separable groups. Several powerful tools for studying residual properties of a group were discovered and developed by the PI in this work. In the first part of the project we intend to use these tools and introduce new ones in order to establish conjugacy separability of many more groups. The second part will be dedicated to investigation of residual finiteness of outer automorphism groups for certain non-positively curved groups. Our approach here will be based on the theorem of Grossman, providing a connection between conjugacy separability of a group G and residual finiteness of Out(G), together with the structure results about automorphisms of relatively hyperbolic groups which were obtained by Bowditch, Levitt and the PI-Osin.

As for any project in the area of Pure Mathematics, main beneficiaries of the proposed research are other mathematicians. However, during the last several years, the objects that we intend to study have been applied to solve important problems in Biology, Robotics and Computer Science. The proposed work is essential for further investigation of these objects and, hence, for their applications. The results of the proposed research will be published in academic journals of international standing. Prior to publication they will be made available on open-access repositories (e.g., www.arxiv.org). The PDRA and the PI will also give talks on conferences and seminars on the topics related to this work. On the other hand, the main goal of this grant proposal is to fund a post-doctoral position for working on the suggested problems. This would provide a unique opportunity for the PDRA to learn the novel theory and tools outlined in the Case for Support, and to work in collaboration with some of the top experts in the field. Such an experience would certainly be invaluable at the later stage of his/her career, and would help in establishing him/herself as a member of the mathematical community in the UK and internationally. Reciprocally, the PDRA would have a positive impact on the Pure Group at the School of Mathematics in the University of Southampton, enriching the Group academically and contributing to the overall scientific environment. The Group has a great track record in hosting post-doctoral researchers and provides an excellent environment for the professional development of young mathematicians. Finally, this grant will support the PI in establishing himself as a new member of the School of Mathematics at the University of Southampton and of the UK's academic community.