| In geometric terms, a Coxeter group is one generated by reflections. They were defined by Jacques Tits in honor of H. S. M. Coxeter. Coxeter groups are very special among groups, since they arise as crucial auxiliary objects in so many different circumstances and branches of mathematics.;We prove that an infinite Coxeter group has a subgroup of finite index which can be mapped homomorphically onto Z. This property is stronger than not having Kazhdan's property (T), and it is an important property for a group to have. In geometric terms, if a group G with this property is the fundamental group of some space X, then X admits a finite covering with a non-zero first Betti number.;There is a standard way of producing an epimorphism from the fundamental group of a manifold onto Z : we consider a two-sided, non-separating submanifold of codimension one, and assign to each loop the sum of its intersection numbers with this submanifold. The main idea of our proof is to try to transpose what this situation gives at the level of the universal cover, to the context of the Coxeter complex associated to a Coxeter group. The Coxeter complex is not very far from the simplicial analogue of a manifold. Its walls are codimension one subcomplexes that under some conditions give "two-sided", "non-separating" subcomplexes of codimension one in the quotient of the Coxeter complex by a subgroup of finite index of the given Coxeter group. In order to make the presentation as clear and as simple as possible, we will alternate geometric and algebraic points of view.;The second result states that an infinite Coxeter group that is not virtually abelian has a finite index subgroup which can be mapped onto a non-abelian free group. The idea of the proof is to consider an action of a finite index normal subgroup on a tree. |
