1. OVERVIEW 3 Taken together, Theorem 1.1 and Theorem 1.3 inductively provide the following target for virtual specialness – a malnormal variant of our main result in Theorem 1.7. Theorem 1.4 (Malnormal Quasiconvex Hierarchy). Suppose G has a malnormal quasiconvex hierarchy. Then G is virtually compact special. Cubical Small-cancellation Theory: A presentation ⟨a,b,... ∣ W1,...,Wr⟩ is C′( 1 n ) if for any “piece” P (i.e. a subword that occurs in two or more ways among the relators) in a relator Wi we have ∣P∣ 1 n ∣Wi∣. For n ≥ 6 the group G of the presentation is hyperbolic and disk diagram methods provide a very explicit understanding of many properties of G. The presentation above can be reinterpreted as ⟨X ∣ Y1,...,Yr⟩ where X is a bouquet of loops and each Yi → X is an immersed circle corresponding to Wi, and the group G of the presentation is then π1X/⟨⟨π1Y1 ...,π1Yr⟩⟩. We generalize this to a setting where X is a nonpositively curved cube complex and each Yi → X is a local isometry. We also offer a notion of C′( 1 n ) small-cancellation theory for such “cubical presentations”. The main results of classical small-cancellation theory – Greendlinger’s lemma and the ladder theorem (and other results involving annular diagrams) have quite explicit generalizations. In particular, we obtain the following result which generalizes the classification of finite trees: T is either a single vertex, is a subdivided arc, or has three or more leaves: Theorem 1.5. If D is a reduced diagram in a cubical C′( 1 24 ) presenta- tion then either D is a single 0-cell or cone-cell, or D is a “ladder” consisting of a sequence of cone-cells, or D has at least three spurs and/or cornsquares and/or shells. The undefined terms in Theorem 1.5 are described in Chapter 9, but the reader might wish to take a glimpse at Figure 9.5. One motivation for introducing a cubical small-cancellation theory is that when the “relators” Yi also have given wallspace structures, then there are natural walls – and hence usually codimension-1 subgroups – in the group G, generalizing the same phenomenon for C′( 1 6 ) groups. This cubical small-cancellation theory helps to coordinate the proof of the following result: Theorem 1.6 (Malnormal Special Quotient Theorem). Let G be hyper- bolic and virtually compact special. Let {H1,...,Hr} be an almost malnor- mal collection of subgroups. There exist finite index subgroups H′,...,H′ 1 r such that G/⟨⟨H′,...,H′⟩⟩ 1 r is virtually compact special and hyperbolic. Most of our exposition circulates around the proof of Theorem 1.6. As- suming that G = π1X, we first choose a collection of local isometries Yi → X with π1Yi = Hi. We then choose appropriate finite covers ̂ i (the H′ i will be π1 ̂ i ) such that the group ¯ of ⟨X ∣ ̂ 1 ,..., ̂ r ⟩ has a finite index subgroup ¯′ with a malnormal quasiconvex hierarchy (we have hidden a few steps

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