Lecture 6

Isoperimetric Function. Two finite presentations P_{1}, P_{n} yield a isomorphic groups ⇔ there is a finite sequence of Tietze transformations from one to another

P_{1} → P_{2} → … → P_{n}.

Tietze transformations:

I. Addition of a new generator:

⟨a_{1}, …, a_{k}|R_{1}, …, R_{s}⟩ → ⟨a_{1}, …, a_{k}, b|R_{1}, …, R_{s}, b = A⟩,

where A is word in a_{i}.

Exercise. Prove that obvious map is isomorphism. II. Addition of a new generator:

⟨a_{1}, …, a_{k}|R_{1}, …, R_{s}⟩ → ⟨a_{1}, …, a_{k}|R_{1}, …, R_{s}, V ⟩,

where V ∈ ⟨⟨R_{i}⟩⟩. Topologically (see figure):

¡figure¿ Exercise. Prove that.

Definition 1. Functions f, g : ℕ → ℕ are equivalent f ∼ g, if ∃K such that

\begin{matrix}f(n) ≤ Kg(Kn) + Kn, \\ g(n) ≤ Kf(Kn) + Kn. \end{matrix}

Theorem 1. Let X, Y be finite 2-complexes with π_{1}X ≅ π_{1}Y. Then f_{X} ∼ f_{Y}.

Proof. It suffices to prove the theorem in case of Tietze transformation.

1) Suppose X →^{I} Y.

a) f_{X}(n) ≤^{?} f_{Y} (n). Let P → X with |P | ≤ n, let D → Y be a disc diagram for P in Y.

¡figure¿ How to remove 2-cell b = A? ¡figure¿ Since P doesn’t pass through b, any minimal area diagram D → Y for P cannot

contain b, or b = A, thus D maps to X, so f_{X}(n) ≤ f_{Y} (n).

b) f_{X}(n) ≥^{?} f_{Y} (n). Let P → Y be a null-homotopic path. Replace each b and P by A to obtain P′ → X. Note that |P′| ≤ |A||P |. Let D′ be a minimal disc diagram for P′ in X. Extend this to diagram D → Y by attaching at most |P | copies of 2-cell b = A.

Area (D) ≤ Area (D′) + |P | ≤
≤ f_{X}(|P′|) + |P | ≤
≤ f_{X}(|A| · |P |) + |P |.

¡figure¿

2) Suppose X ⟶^{II} Y.

¡figure¿

a) f_{X}(n) ≤^{?} f_{Y} (n). Let E be a disc diagram in X whose boundary path is ∂V. ¡figure¿ Claim: f_{X}(n) ≤ Area (E) f_{Y} (n). Proof. Let P → X be null-homotopic. Let D → Y be a minimal disc diagram

for P. Substitute E for each occurrence of V in D to obtain new diagram D′ → Y.

¡figure¿ Area (D′) ≤ Area (E) Area (D) (what if it’s singular?).

b) f_{X}(n) ≥ f_{Y} (n) obviously.

□

Now we have a kind of a group invariant.

Definition 2. Group word-hyperbolic if it has a finite presentation whose standard 2-complex X has linear isoperimetric function: ∃K such that f_{X}(n) ≤ Kn.

Note. f(n) ≤ \frac{1}{100}n^{2} ⇒ ∃K such that f(n) ≤ Kn. That means there is a gap between quadratic and linear functions in the collection of all isoperimetric function.

Construction. Let X be a complex that is a graph of spaces (see figure) with each vertex space a subcomplex and each edge space a (subcomplex) × I.

¡figure¿ Let Φ : Y → X be combinatorial map of complexes. Then Y has an induced

structure as a graph of spaces. Indeed, the vertex spaces of Y are are components of Φ^{−1}(X_{V}), where X_{V} is vertex space of X. Open edge spaces are components of Φ^{−1}(X_{e} × (−1, 1)).

Example. Subcomplex of X is a graph of spaces. Cover of X decomposes as a graph of spaces. Universal cover of X is a tree of spaces.

¡figure¿ Covering space K(2, 3) × ◯ (see figure) ¡figure¿

ℤ \begin{matrix}∗ \\ ℤ \end{matrix} ℤ ≅ ⟨c, b|c^{3} = b^{2}⟩ ⟵^{index 6} 𝔽_{2} × ℤ ≅ π_{1}(K(2, 3) × ◯).

Universal cover (see figure): ¡figure¿ ¡figure¿ π_{1}X acts on X̃ and thus acts on ΓX̃