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This project which was funded by the ANR for the period 2012-2016,
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 proofs of Thurston's geometrization conjecture as well as of the "virtually Haken" and "virtual fibration" conjectures, have considerably sharpened our understanding of 3-manifolds. The latter two results state that a hyperbolic 3-manifold admits a finite sheeted covering containing an embedded essential surface, and, even better, a finite sheeted covering admitting a surface fibration over the circle. Their proofs are based on tools coming from both geometric group theory and low-dimensional topology. In particular, they rely on a deep understanding of the subgroups of the fundamental groups of hyperbolic 3-manifolds.

Nonetheless, a major question concerning the structure of hyperbolic 3-manifolds and their fundamental groups is still open: this is Cannon's conjecture which proposes a dynamical characterisation of the fundamental groups of hyperbolic 3-manifolds. One way to attack this conjecture is to look for splittings of the fundamental groups into elementary blocks.

In a more algebraic setting, Gromov's conjecture also deals with the study of subgroups inside hyperbolic groups and concerns the existence of surface subgroups inside hyperbolic (non-elementary and non free products) groups.

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 Agol, Wise et al. and provide the main ingredient in the proof of the virtual Haken and fibration conjectures.

-Lp-cohomology. Lp-cohomology was exploited by Bourdon to detect splittings of hyperbolic groups.

-Dynamics of the induced action on the boundary. Note that this also relevant to the boundary characterisation of Kleinian groups, and Cannon's conjecture.

In this way, we hope 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.

The ANR will provide a**12 month Post-Doc fellowship** for a mathematician
working in the fields touched upon by the project. Please follow this
link for more details about the position.

In recent years, the proofs of Thurston's geometrization conjecture as well as of the "virtually Haken" and "virtual fibration" conjectures, have considerably sharpened our understanding of 3-manifolds. The latter two results state that a hyperbolic 3-manifold admits a finite sheeted covering containing an embedded essential surface, and, even better, a finite sheeted covering admitting a surface fibration over the circle. Their proofs are based on tools coming from both geometric group theory and low-dimensional topology. In particular, they rely on a deep understanding of the subgroups of the fundamental groups of hyperbolic 3-manifolds.

Nonetheless, a major question concerning the structure of hyperbolic 3-manifolds and their fundamental groups is still open: this is Cannon's conjecture which proposes a dynamical characterisation of the fundamental groups of hyperbolic 3-manifolds. One way to attack this conjecture is to look for splittings of the fundamental groups into elementary blocks.

In a more algebraic setting, Gromov's conjecture also deals with the study of subgroups inside hyperbolic groups and concerns the existence of surface subgroups inside hyperbolic (non-elementary and non free products) groups.

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 Agol, Wise et al. and provide the main ingredient in the proof of the virtual Haken and fibration conjectures.

-Lp-cohomology. Lp-cohomology was exploited by Bourdon to detect splittings of hyperbolic groups.

-Dynamics of the induced action on the boundary. Note that this also relevant to the boundary characterisation of Kleinian groups, and Cannon's conjecture.

In this way, we hope 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.

The ANR will provide a