Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 133 / 1 / a

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- Solution a

Solution

Put

u = (1021), v = (1201) .

For every nonzero integer

n

,

u^{n} = (10 2 n 1), v^{n} = (1 2 n 01) .

Let

A = {(x, y) : ∣ x ∣ > ∣ y ∣}, B = {(x, y) : ∣ x ∣ < ∣ y ∣} .

These are disjoint nonempty subsets of

R^{2}

. If

(x, y) \in B

, then

∣ x + 2 n y ∣ \geq 2∣ n ∣∣ y ∣ - ∣ x ∣ > ∣ y ∣,

so

u^{n} (B) \subseteq A

. Similarly, if

(x, y) \in A

, then

∣2 n x + y ∣ \geq 2∣ n ∣∣ x ∣ - ∣ y ∣ > ∣ x ∣,

so

v^{n} (A) \subseteq B

.

The ping-pong lemma now identifies the subgroup generated by

u

and

v

with

⟨ u ⟩ * ⟨ v ⟩ ≅ Z * Z ≅ F_{2} .

Both generators have infinite order, so this is a free group of rank two inside the special linear group

SL_{2} (Z)

.

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