The Farey graph, continued fractions, and the modular...

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The Farey graph, continued fractions, and themodular group

Ian Short

26 November 2009

Collaborators

Alan Beardon (University of Cambridge)

Meira Hockman (University of the Witswatersrand)

Continued fractions

Euclid’s algorithm

31

13

= 2 +1135

= 2 +1

2 +153

= 2 +1

2 +1

1 +132

31

13= 2 +

1135

= 2 +1

2 +153

= 2 +1

2 +1

1 +132

31

13= 2 +

1135

= 2 +1

2 +153

= 2 +1

2 +1

1 +132

31

13= 2 +

1135

= 2 +1

2 +153

= 2 +1

2 +1

1 +132

31

13= 2 +

1

2 +1

1 +1

2

The nearest-integer algorithm

31

13

= 2 +1135

= 2 +1

3 +1

−52

= 2 +1

3 +1

−3 +1

2

31

13= 2 +

1135

= 2 +1

3 +1

−52

= 2 +1

3 +1

−3 +1

2

31

13= 2 +

1135

= 2 +1

3 +1

−52

= 2 +1

3 +1

−3 +1

2

31

13= 2 +

1135

= 2 +1

3 +1

−52

= 2 +1

3 +1

−3 +1

2

31

13= 2 +

1

3 +1

−3 +1

2

Another expansion

31

13= 3 +

1

−2 +1

3 +1

−3

Questions

Is the nearest-integer continued fraction the shortest continuedfraction with integer coefficients?

How many shortest continued fractions are there?

Can we characterise shortest continued fractions?

Questions

Literature

Perron O., Die Lehre von den Kettenbruchen, ChelseaPublishing Company, New York, (1950).

Srinivisan, M.S., Shortest semiregular continued fractions, Proc.Indian Acad. Sci., Sect. A, 35 (1952).

Literature

The modular group

Mobius transformations

Define, for an integer b,

Sb(z) = b + 1/z.

Sb1 ◦ Sb2 ◦ · · · ◦ Sbn(∞) = b1 +1

b2 +1

b3 +1

b4 + · · ·+1

bn

Sb(z) = b + 1/z.

Sb1 ◦ Sb2 ◦ · · · ◦ Sbn(∞) = b1 +1

b2 +1

b3 +1

b4 + · · ·+1

bn

Sb(z) = b + 1/z.

Sb1 ◦ Sb2 ◦ · · · ◦ Sbn(∞) = b1 +1

b2 +1

b3 +1

b4 + · · ·+1

bn

Note that Sbi(0) =∞.

Hence

Sb1 ◦ Sb2 ◦ · · · ◦ Sbn(0) = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn−1(∞).

Note that Sbi(0) =∞. Hence

Sb1 ◦ Sb2 ◦ · · · ◦ Sbn(0) = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn−1(∞).

What is the group generated by the maps Sbi(z) = bi + 1/z?

(b 11 0

)7−→ Sb

What is the group generated by the maps Sbi(z) = bi + 1/z?(b 11 0

)7−→ Sb

The modular group

Γ =

{z 7→ az + b

cz + d: a, b, c, d ∈ Z, ad− bc = 1

}

T (z) = z + 1 X(z) = −1/z

The modular surface

The extended modular group

Γ =

{z 7→ az + b

cz + d: a, b, c, d ∈ Z, |ad− bc| = 1

}

Presentation for the extended modular group

T (z) = z + 1 U(z) =1

zV (z) = −z

V = UTUT−1UT

Γ =⟨T,U | U2 = 1, UV = V U, V T = T−1V

⟩

T (z) = z + 1 U(z) =1

zV (z) = −z

V = UTUT−1UT

Γ =⟨T,U | U2 = 1, UV = V U, V T = T−1V

⟩

T (z) = z + 1 U(z) =1

zV (z) = −z

V = UTUT−1UT

Γ =⟨T,U | U2 = 1, UV = V U, V T = T−1V

⟩

Words in the extended modular group

Each word in Γ has the form

T b1UT b2U · · ·T bnU,

for integers b1, b2, . . . , bn (T (z) = z + 1 and U(z) = 1/z).

Sb = T bU

Sb1Sb2 · · ·Sbn ,

for integers b1, b2, . . . , bn.

Sb = T bU

A correspondence

Finite integer continued fractions

Words of T and U in Γ

(T (z) = z + 1 and U(z) = 1/z)

A correspondence

(T (z) = z + 1 and U(z) = 1/z)

A correspondence

(T (z) = z + 1 and U(z) = 1/z)

A correspondence

(T (z) = z + 1 and U(z) = 1/z)

The word metric

Cayley graph of Γ with respect to T (z) = z + 1 and X(z) = −1/z

The Farey graph

The upper half-plane

The vertical half-plane

A path of geodesics

Tn = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn

A path of geodesics

Tn = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn

A path of geodesics

Tn = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn

A path of geodesics

Tn = Sb1 ◦ Sb2 ◦ · · · ◦ Sbn

The Farey graph

Γ(δ)

The Farey graph

Vertices = Q

Join ab to c

d if and only if |ad− bc| = 1.

The Farey graph

Vertices = Q

Join ab to c

The Farey graph

Vertices = Q

Join ab to c

The Farey graph

Mediants

The Farey graph

Paths in the Farey graph

A correspondence

Finite paths from ∞ in the Farey graph

A correspondence

Cutting sequences

Geodesics on the modular surface

A geodesic on the modular surface

Geodesics

Questions

Is the nearest-integer continued fraction the shortest continuedfraction?Does the nearest-integer continued fraction correspond to ageodesic in the Farey graph?

How many shortest continued fractions are there?How many geodesics are there between two vertices in the Fareygraph?

Can we characterise shortest continued fractions?Can we characterise geodesics in terms of the coefficients of acontinued fraction?

Questions

Is the nearest-integer continued fraction the shortest continuedfraction?

Does the nearest-integer continued fraction correspond to ageodesic in the Farey graph?

Questions

How many geodesics are there between two vertices in the Fareygraph?

Questions

Can we characterise geodesics in terms of the coefficients of acontinued fraction?

Questions

Answer 1

The nearest-integer algorithm does correspond to a geodesic.

Answer 1

The nearest-integer algorithm does correspond to a geodesic.

Answer 2

Count geodesics using the dual graph.

Answer 2

Count geodesics using the dual graph.

The dual graph

Unique path of triangles

Answer 3

Theorem. A continued fraction with coefficients b1, b2, . . . , bncorresponds to a geodesic in the Farey graph if and only if|bi| > 2 for each i > 2, and there is no substring of b2, b3, . . . , bnof the form 2,−3, 3,−3, 3,−3, . . . , 3,−2.

Answer 3

Theorem. A continued fraction with coefficients b1, b2, . . . , bncorresponds to a geodesic in the Farey graph if and only if|bi| > 2 for each i > 2, and there is no substring of b2, b3, . . . , bnof the form 2,−3, 3,−3, 3,−3, . . . , 3,−2.