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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