Lecture 5

Disc Diagrams.

Definition 1. A disc diagram is a finite contractible planar 2-complex D.

¡figure¿ We usually think of D as having a particular planar embedding D ⊂ 𝔼^{2}. (Indeed,

attaching a 2-cell along the border of a 2-complex we obtain such an embedding.)

A singular point of D is a point where is not locally a surface. Given an embedding D ⊂ 𝔼^{2}, set of all singular points is a topological boundary of D.

Singular 1-cells are divided into two types:

isolated 1-cells, which have no 2-cells attached along, and boundary 1-cells = singular\ isolated.

For example, the edge in the figure is not boundary, having one 2-cell attached twice.

¡figure¿ Choose a base point b ∈ ∂D. The boundary cycle path P of D based at b is

a combinatorial path traveling “around D” starting and ending at b, that passes through each boundary edge and through each isolated edge twice.

P corresponds to attaching map of the 2-cell, whose interior is 𝔼^{2}\D Some notations:

Area (D) = number of 2-cells in D.

∂D is considered path in D.

D is none-singular, if D is homeomorphic to B^{2}, i.e. has no singular

points. |∂_{P} D| = length of boundary path. |P | = length of path P.

Definition 2. A disc diagram in a 2-complex X is a combinatorial map D → X.

Combinatorial map Y → X between complexes sends open cells homeomorphically to open cells (of some dimension).

Convention: our 2-complexes are combinatorial, and attaching maps of 2-cells are combinatorial (after appropriate subdivision). That is, X is X′ with polygons attached to it (in general case, attaching maps could be horrible, see figure). However, here we are missing “a disk attached to 0-cell”.

¡figure¿

Theorem 1. Let P → X be a closed combinatorial path. TFAE:

1) P is null-homotopic.

2) P → X factors as P → D → X where D → X is a disc diagram and P → D is a boundary path.

¡figure¿ In fact, this (see figure) can be made into proof. Proof. 2⇒ 1 obviously.

\xymatrix{π_{1}P \ar[dr] \ar[rr] &  & π_{1}X \\  & π_{1}D \ar[ur]}

1⇒ 2

We assume X^{0} = {·}, so X is standard 2-complex of a presentation ⟨g_{1}, … |R_{1}, …⟩. Also, we assume π_{1}X ≅ G of this presentation (isomorphism is induced by obvious map).

The path is null-homotopic ⇔ P ∼ 1 in π_{1}X ⇔ the word P corresponding to path P represents 1 in G ⇔ P is freely equivalent to a product relators and inverses, p ∼ c_{1}^{−1}S_{1}^{±1}c_{1}c_{2}^{−1}S_{2}^{±1}c_{2} ⋯ c_{n}^{−1}S_{n}^{±1}c_{n} = P_{0} (c_{i} are words, S_{i} are some relators R_{j}).

A trick: ¡figure¿

∂_{P} D_{0} = ∏ c_{i}^{−1}S_{i}^{±1}c_{i} ∼ P D_{0} obviously maps to X (according to labels on its edges).

Removal of backtracks. There is a sequence of words P_{0}, P_{1}, …, P_{t}, P_{i+1} is obtained from P_{i} by insertion or deletion of a backtrack. We follow this with a corresponding sequence of disc diagrams D_{0}, D_{1}, …, D_{t} = D, D_{i+1} is obtained from D_{i} by folding or removing spurs.

(A spur is “isolated” 1-cell with a “pendant” end.)

¡figure¿

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Example. ⟨a, b|bab^{−1}a^{−1}⟩ Applying this argument we can see that (see figure):

bbaabab^{−1}a^{−1}b^{−1}b^{−1}a^{−1}a^{−1} ∼ 1 ∼
b(bab^{−1}a^{−1})b^{−1} · baba(bab^{−1}a^{−1})(baba)^{−1} · ba(bab^{−1}a^{−1})(ba)^{−1}·
bab^{−1}a^{−1} · a(bab^{−1}a^{−1})a^{−1}

¡figure¿

This process can be inversed: given a disc diagram D for p (meaning with boundary graph p) we can easily express p as a freely equivalent to a product of conjugate relators.

¡figure¿

If we are careful, the length of conjugators c_{i} are bounded by 2 ∑ |∂R_{i}| + |∂_{p}D|, where R_{1}, …, R_{n} are in the 2-skeleth of D.

¡figure¿

Isoperimetric Function.

Definition 3. Given a null-homotopic path p → X, define

Area (p) = inf{Area (D) |p = ∂_{p}D}

Definition 4. Given a 2-complex X, its isoperimetric function f : ℕ → ℕ (here, 0, ∞ ∈ ℕ) is:

f(n) = sup{Area (p) |p → X nullhomotopic, |p| ≤ n},

or f(n) = largest area needed to fill in null-homotopic path of length ≤ n.

Define isoperimetric function of presentation in terms of standard 2-complex.

Example.

⟨a, b|−⟩ f(n) = 0

⟨a, b|ab = ba⟩ f(n) ≤ \frac{n^{2}}{4} , f(n) ∼ n^{2}
⟨a, b|ab = bba⟩ f(n) ≤ 2^{n}, f(n) ∼ 2^{n}

The quantitative nature of isoperimetric function of a finitely presented group 1) doesn’t really depend on choice of finite presentation,

2) tells us a great deal about the group.

Theorem 2. ⟨a_{1}, …, a_{n}|R_{1}, …, R_{n}⟩ has a solvable word problem ⇔ isoperimetric function is computable (recursive).

Proof. Solvable word problem ⇒ f(n) is recursive. Enumerate all P such that |P | ≤ n and P represents 1_{G}; find smallest disc

diagram, which represent path p.

Solvable word problem ⇐ f(n) is recursive. Given word P, compute f(|P |). Build all disc diagrams in X with ≤ f(|P |) 2-

cells and boundary path p. If P = 1_{G}, you will find some D_{j}, p = ∂_{p}D_{j}, otherwise P ≠ 1.

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