Reflection subgroups of Coxeter groups

We use the geometry of the Davis complex of a Coxeter group to investigate finite index reflection subgroups of Coxeter groups. The main result is the following: if $G$ is an infinite indecomposable Coxeter group and $H\subset G$ is a finite index reflection subgroup, then the rank of $H$ is not less than the rank of $G$. This generalizes earlier results of the authors (2004). We also describe the relationship between the nerves of the group and the subgroup in the case of equal rank.