Lecture 12

Small Cancellation Theory.

Definition 1. Let D → X be a disk diagram in some 2-complex. A cancelable pair of 2-cells in D is a pair of 2-cells R_{1}, R_{2} meeting along a 1-cell e in D such that ∂R_{1}, ∂R_{2} starting at e project to the same path in X.

For instance, if R_{1}, R_{2} fold together along the way to X, the R_{1}, R_{2} are the cancelable pair.

(figure: a near-immersion with a cancelable pair)

When a presentation has duplicate relators (e.g. ⟨a, b|abb, abb, babaa⟩) it is natural to have near-immersions with cancelable pairs. With X = ⟨a, b|(ab)^{3}⟩, X̃ has many duplicate 2-cells.

(figure: tree of hexagons)

Exercise. Show X has an index 6 subgroup that is free (degree 6 covering map).

Definition 2. A disc diagram D → X is reduced if it has no cancelable pairs.

Note. If X has no duplicate 2-cells, no 2-cells with periodic attaching maps, then disc diagrams in X are reduced iff D → X is a near-immersion.

Exercise. If a disc diagram satisfies negative weight test, then it has no cancelable pairs.

Note. Any minimal area disc diagram with boundary path p is reduced, since we can remove cancelable pairs to decrease area. Converse does not hold. Relationship in 2 dimensions is not understood, counterexamples are not aspherical. (figure: cubes)

Small Cancelation Conditions. The maximal internal arcs in reduced disc diagrams are called pieces.

Definition 3. X satisfies C(p) condition if for any reduced disc diagram D → X no boundary path of a 2-cell R in D is concatenation of < p pieces.

In other words, reduced daisy with less than p petals doesn’t exist.

(figure: daisy flower)

If X is C(p) then in any reduced disc diagram D → X when we ignore valence 2 vertices in D, each 2-cell of D without boundary (lying in ∂D) 1-cells looks like ≥ p-gon.

(figure: rectangular figure) X might be C(4), not C(5).

Definition 4. X satisfies T (q) condition if each reduced disc diagram D → X has the property thattach internal 0-cell v ∈ D satisfies either deg v = 2 or deg v ≥ q.

(figure: pizza pie) Equivalently, any reduced pizza pie → X has 2 slices, or ≥ q slices.

These definitions usually presume that attaching maps of 2-cells are immersions. Example. ⟨a, b|abb^{−1}⟩ (figure: one-arrowed clock) T (2) — attaching maps are immersions. Note. T (q) implies immersed attaching maps for q ≥ 2. We refer to a presentation as C(p) − T (q) if its standard 2-complex satisfies

C(p) − T (q).

Note. A piece in a presentation is a word p that appears in “more than one way” among relators.

Example. ⟨a, b|abbaaabbaaab⟩ — aa, abbaaab. ⟨a|aaaaaaa⟩ is C(∞) − T (∞). For instance, a diagram corresponding to piece

aaa is not reduced. (figure: a diagram)

⟨a, b, c|(abc)^{100}⟩ is C(∞) − T (∞). Exercise. Every group has a C(5) presentation (also has C(4) − T (3) and

C(3) − T (4) presentations). It’s about subdivision.

Definition 5. X satisfies C′(α) condition if for each piece p occurring in ∂_{p}R we have |p| < α|∂_{p}R|. (This is a metric small-cancelation condition.)

Note. C′(\frac{1}{n}) ⇒ C(n + 1). Example. ⟨a, b|abab^{2}ab^{3} ⋯ ab^{100}, baba^{2}ba^{3} ⋯ ba^{100}⟩. A piece is not longer

than b^{98}ab^{99}, so it’s C′(∼ \frac{4}{99}) ⇒ C(6).