  The paper is concerned with Kropholler's conjecture on splitting a finitely
generated group over a codimension-1 subgroup. For a subgroup H of a group G,
we define the notion of "finite splitting height" which generalises the
finite-height property. By considering the dual CAT(0) cube complex associated
to a codimension-1 subgroup H in G, we show that the Kropholler-Roller
conjecture holds when H has finite splitting height in G. Examples of subgroups
of finite height are stable subgroups or more generally strongly quasiconvex
subgroups. Examples of subgroups of finite splitting height include relatively
quasiconvex subgroups of relatively hyperbolic groups with virtually polycyclic
peripheral subgroups. In particular, our results extend Stallings' theorem and
generalise a theorem of Sageev on decomposing a hyperbolic group by quasiconvex
subgroups.
