These are the notes of a course given by the two authors for the 25th Colóquio Brasileiro de Matemática and addressing the problem of the existence of incompressible surfaces inside a (hyperbolic) \(3\)-manifold using group theoretical approaches.
The first Chapter is a brief review (without proofs) of basic notions and results in hyperbolic geometry (hyperbolic structures, models for hyperbolic space, classification of isometries,…) and \(3\)-manifold topology (irreducibility, incompressible surfaces,…) as well as of open conjectures related to the existence of incompressible surfaces inside manifolds (surface conjecture, virtually Haken conjecture, virtually positive Betti number conjecture,…).
In Chapter two, some group theoretical properties are discussed, namely residual finiteness and subgroup separability. A group \(G\) is said to be \(H\)-subgroup separable, for a finitely generated subgroup \(H\), if for all \(g\in G\setminus H\) there exists a finite index subgroup \(K\) which contains \(H\) but not \(g\). Note that a group is residually finite iff it is \(1\)-subgroup separable. A group which is \(H\)-subgroup separable for all finitely generated \(H\) is called subgroup separable or LERF (Locally Extended Residually Finite). Topological characterisations of these notions for fundamental groups of manifolds in terms of properties of finite-sheeted covers are also given.
After remarking that a hyperbolic manifold whose fundamental group is LERF and contains a surface subgroup is virtually Haken, the authors give different sufficient conditions for a discrete subgroup of \(Iso({\mathbb H^n})\) to be subgroup separable, in particular this is the case for Bianchi groups, and for groups generated by the reflections in the faces of a right-angled finite-volume polyhedron (Scott’s theorem). The authors also note that, as a consequence of the Tameness conjecture, the fundamental group of a finite volume hyperbolic \(3\)-manifold is subgroup separable with respect to its finitely generated geometrically infinite subgroups.
The existence of hyperbolic manifolds whose fundamental groups do not contain surface groups shows that one cannot hope to solve the conjectures mentioned above just by understanding the LERF property for fundamental groups. Other group theoretical properties which seem to be more useful in addressing the problem are discussed in Chapter four. Here the concept of virtual retraction is introduced: the existence of a virtual retraction of a fundamental group to any of its cyclic groups (i.e. there exists homomorphism \(\theta:V\longrightarrow H\) where \(H\) is cyclic subgroup and \(V\) a finite index subgroup containing \(H\)) would imply that the manifold has virtually positive Betti number. It is then shown that Coxeter groups, all of whose two-generator special subgroups are finite, virtually retract over a cyclic subgroup acting hyperbolically on the Coxeter complex. In particular, a Coxeter group is either virtually abelian or has infinite virtual Betti number.