2 1. OVERVIEW groups embed in (and are closely allied with) right-angled Coxeter groups, this means that one can obtain linearity and residual finiteness by verifying virtual specialness. We describe some criteria for verifying that a nonpositively curved cube complex is virtually special the most fundamental is the condition that double hyperplane cosets are separable. A deeper criterion [36] arises from a nonpositively curved cube complex that splits along an embedded 2-sided hyperplane into one or two smaller nonpositively curved cube complexes. Under good enough conditions the resulting cube complex is virtually spe- cial: Theorem 1.1 (Specializing Amalgams). Let Q be a compact nonposi- tively curved cube complex with π1Q hyperbolic. Let P be a 2-sided embedded hyperplane in Q such that π1P π1Q is malnormal and each component of Q No(P ) is virtually special. Then Q is virtually special. A subgroup H of G is “codimension-1” if H splits G into two or more “deep components” like an infinite cyclic subgroup of a surface group. In his PhD thesis, Sageev understood that when G acts minimally on a CAT(0) cube complex ̃, the stabilizers of hyperplanes are virtually codimension-1 subgroups of G. He contributed an important converse to this: Construction 1.2 (dual CAT(0) cube complex). Given a group G and a collection H1,...,Hr of codimension-1 subgroups, one obtains an action of G on a dual CAT(0) cube complex whose hyperplane stabilizers are commensurable with conjugates of the Hi. We review Construction 1.2 in the context of Haglund-Paulin wallspaces, and describe some results on the finiteness properties of the action of G on the CAT(0) cube complex ̃. The main point is that if we can produce sufficiently many quasiconvex codimension-1 subgroups in the hyperbolic group G, then we can apply Construction 1.2 to obtain a proper cocompact action of G on a CAT(0) cube complex. This is how we prove the following result [43]: Theorem 1.3 (Cubulating Amalgams). Let G be a hyperbolic group that splits as A∗C B or A∗Ct=D where C is malnormal and quasiconvex. Suppose A,B are each fundamental groups of compact cube complexes. And suppose that some technical conditions hold (and these hold when A,B are virtually special). Then G is the fundamental group of a compact nonpositively curved cube complex. A hierarchy for a group G is a way to repeatedly build it starting with trivial groups (but sometimes other basic pieces) by repeatedly taking amal- gams A ∗C B and A∗Ct=D whose vertex groups have shorter length hierar- chies. The hierarchy is quasiconvex if at each stage the amalgamated sub- group C is a finitely generated that embeds by a quasi-isometric embedding, and similarly, the hierarchy is malnormal if C is malnormal in A ∗C B or A∗Ct=D.
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