Geometry of Subgroups – GDSous/GSG

This project concerns basic research in fundamental mathematics, more specifically it combines geometry in a broad sense, low dimensional topology, group theory, dynamical systems, and geometric analysis.

In recent years, the proof of Thurston's geometrization conjecture has considerably sharpened our understanding of 3-manifolds. Nevertheless, three main questions concerning the structure of a hyperbolic 3-manifold and its fundamental group still remain unanswered.

The first two questions, known as the « virtually Haken » and « virtual fibration » conjectures, address the problem of understanding whether any given hyperbolic 3-manifold admits a finite cover containing an essential surface or, even better, admitting a surface fibration over the circle; the third one, Cannon's conjecture, deals with the dynamical characterisation of its fundamental group.

These questions can be restated in more algebraic terms. For the first two, this boils down to asking whether a Kleinian group (cocompact or with finite covolume) contains a surface group (i.e. the manifold contains an immersed surface) enjoying certain separability properties (i.e. the immersion lifts to an embedding in some finite cover of the manifold) and possibly having specified limit set (i.e. the surface is virtually a fibre of a fibration over the circle). The third one can be tackled by looking for splittings of the fundamental group into elementary bricks.

Motivated by the above questions that show the importance of understanding the subgroup structure of certain groups as well as their splittings, the aim of the present project is to develop techniques to detect the existence of special subgroups (surface subgroups, quasiconvex subgroups) in groups that are relevant in geometry or dynamics (hyperbolic and relatively hyperbolic groups, CAT(0) groups, and convergence groups) and to study their properties or to establish conditions under which certain properties hold. We note that the understanding of the subgroup structure of a group is strongly connected to the understanding of the splittings of the group.

The main tools we wish to exploit to tackle these problems are:
- Cubulations. We remark that cubulations have been exploited by Wise et al. and provide a setting in which the virtual fibration conjecture holds.
- Lp-cohomology. Lp-cohomology was exploited by Bourdon in particular to detect splittings of hyperbolic groups.
- Dynamics of the induced action on the boundary. Note that this is also relevant to the boundary characterisation of Kleinian groups and to Cannon's conjecture.

Although proving these conjectures appears to be an extremely ambitious task, the hope is that the results obtained while carrying out this project will provide useful insight on the subject. Also, one can expect to understand whether results which are known to be valid for hyperbolic 3-manifolds extend to more general settings (hyperbolic and relatively hyperbolic groups, CAT(0) groups).

One of the strong points of the project is that it brings together several mathematicians working in different fields, all relevant to the project, namely combinatorial and geometric group theory, low dimensional topology, hyperbolic geometry, conformal dynamics, etc. Members of the project are also among the authors of some important recent breakthroughs in the subject. A feature of the project will be to encourage members to share their specific expertise and acquire new knowledge at the same time.

The project will be structured by biannual meetings having the three following purposes: providing background on each topic, presenting the main open questions and advances obtained, and setting up collaborations on the future tasks.

This will be one of the scopes of the « ateliers » that will be organised twice a year.