Lecture 8

Theorem 1. For a compact angled 2-complex X, 2πχ(X) = ∑ κ(v) + ∑ κ(f)

Proof. Tautology. Let C = ∑ ∠(c)_{c∈corners(X)}.

Let P = ∑ #e_{e∈1-cells(X)}, where #e = total number of 2-cells appearing locally along

e = times e appears in attaching maps of 2-cells = the number of pages in the book spined by e.

(figure: a book) Example. ⟨a, b|ab^{−1}aab⟩ — #a = 3, #b = 2. We will show:

(1) ∑ κ(f) = C − πP + 2πF,

(2) ∑ κ(v) = 2πV − 2πE + πP − C,
so ∑ κ(f) + ∑ κ(v) = 2πV − 2πE + 2πF = 2πχ(X),

where V is number of 0-cells, E is number of 1-cells, F is number of 2-cells in X. Proof of (1).

κ(f) = ∑ ∠(c)_{c∈corners(f)} −|f|π + 2π
∑_{f∈X} κ(f) = ∑_{f} ∑ ∠(c)_{c∈corners(f)} − ∑_{f} |f|2π + ∑_{f} 2π
∑_{f∈X} κ(f) = C − πP + 2πF

Proof of (2).

κ(v) =  2π −  πχ(link(v))  − ∑ ∠(c)_{c∈corners(v)}
∑_{v∈X} κ(v) = ∑_{v∈X} 2π −  ∑_{v∈X} πχ(link(v))  − ∑_{v∈X} ∑ ∠(c)_{c∈corners(v)} =
=  2πV − ∑_{v∈X} π · (\begin{matrix}vertices \\ in link(v) \end{matrix}) + ∑_{v∈X} π · (\begin{matrix}edges \\ in link(v) \end{matrix}) −  C  =
=  2πV −  2πE + πP  −  C
□

Example. (figure: a graph) If 2-complex is, actually, a graph (i.e. has no 2-cells), we have (for graph in the

figure)

κ(v) = 2π − π deg(v)
2πχ(X) = −4π
χ(X) = −2 = 7 − 9

Example. (figure: an earring squared (with right angles)) We have one 0-cell, four 1-cells, four 2-cells. χ(X) = χ(earring) · χ(earring) =

(−1) · (−1) = 1.

Compute curvatures: κ(f) = 0,

κ(v) = 2π − πχ(link(v)) − ∑ ∠ = 2π + 8π − \frac{16}{2}π = 2π.

(We have link(v) = K(4, 4), so χ(link(v)) = 8 − 16 = −8.) So 2πχ(X) = 2π, i.e. χ(X) = 1.

Nonpositive and negative (conformal) curviture.

Definition 1. (Gersten-Pride) An angled 2-complex satisfies the nonpositive weight test if

1) κ(f) ≤ 0 for each 2-cell f,

2) for each o-cell v ∈ X^{0} and each closed nontrivial immersed path σ in link(v) with σ = e_{1}e_{2} … e_{n},

|σ|_{∠} = ∑_{i=1}^{l} ∠(e_{i}) ≥ 2π,

and the negative weight test if both inequalities are strict.

(figure: example)

Remark. If all angles in X are nonnegative, it suffices to consider simple cycles in the (2) of definition. Indeed, any closed immersed path can be expressed as σ = σ_{1}σ_{2} … σ_{k}, where each σ_{i} is simple and |σ|_{∠} = ∑_{i=1}^{k} |σ_{i}|_{∠} ≥ k2π ≥ 2π.

(figure: a dumbbell) This example shows that requirement of nonnegativeness is essential.

Definition 2. The girth of a graph Γ is the length |σ| of the shortest simple cycle |σ|. If Γ is a forest, girth(Γ) = ∞.

Example. girth(cube) = 4, girth((figire: triangle with a double side)) = 2, girth((figure: dumbbell)) = 1.

Definition 3. A (p, q)-complex X is a 2-complex satisfying 1) each 2-cell has |f| ≥ p,

2) for each 0-cell v ∈ X^{0}, link(v) has girth ≥ q.

(figure: square tiling) — (4,4), also (3,3), (1,4),…

(figure: hexagon tiling) — (6,3)

(figure: trigon tiling) — (6,3)

(figure: cube) — (4,3)

In a sense, the (p, q) condition is a combinatorial comparison condition with a regular tiling of type (p, q).

Lemma 1. If \frac{1}{p} + \frac{1}{q} ≤ \frac{1}{2}, then the (p, q-complex has an angle assignment satisfying nonpositive weight test. Moreover, if \frac{1}{p} + \frac{1}{q} < \frac{1}{2}, then it satisfies negative weight test.

Proof. Assign angle of \frac{p−2}{2}π to each corner. Then

κ(f) = |f| · \frac{p − 2}{p}π − (|f| − 2)π = 2π − \frac{|f|}{p}2π ≤ 0,

and < 0 if |f| > p. Since \frac{1}{p} + \frac{1}{q} ≤ \frac{1}{2} implies \frac{2}{q} ≤ 1 − \frac{2}{p}, for each simple cycle in the link(v) we have

|σ|_{∠} ≥ q · \frac{p − 2}{p} · π = q (1 − \frac{2}{p}) π ≥ 2π,

and > 2π if \frac{1}{p} + \frac{1}{q} < \frac{1}{2}.

Angle assignment \frac{2π}{q} works equally well. If we take average of these angle assignments, then κ(f) < 0, κ(v) < 0, so we obtain negative weight test.

□

Now we have an enormous supply of (p, q)-complexes. Remark. If X is compact, then there are finitely many real numbers arising

as angles, then if always κ(f) < 0 and |σ|_{∠} ≥ 2π, increasing all angles slightly, if always κ(f) ≤ 0 and |σ|_{∠} > 2π, decreasing all angles slightly,

we obtain negative weight test.

Example. ⟨a, t|a^{t} = a^{2}⟩ (figure: standard 2-complex and link) Exercise. G = ⟨a_{1}, a_{2}, …, a_{n}, t|(A_{1})^{t} = A_{2}⟩ where A_{1}, A_{2} are freely reduced

nonempty words in a_{i}^{±1}. Verify that standard 2-complex satisfies nonpositive weight test.

Example. ⟨a, b|aaabbbabab^{−1}ab^{−1}⟩. (figure: link) Standard 2-complex doesn’t satisfy the weight test, since relator contains each

pair of generators twice and therefore the link is a full 4-graph with double edges and κ(f) ≥ 6 · 2π − 10π ≥ 2π > 0.

Exercise. Find a 2-complex with exactly one 2-cell such that it has a unique angle assignment satisfying the weight test.