Farey graph

In mathematics , the Farey graph is an infinite graph that has numerous applications in number theory and other areas of mathematics.

definition

The set of nodes in the Farey graph is the set of all pairs ${\ displaystyle \ mathbb {Q} \ cup \ left \ {\ infty \ right \}}$

{\ displaystyle \ left \ {{\ frac {p} {q}}: p, q \ in \ mathbb {Z} \ {\ mbox {coprime}}, q \ geq 0 \ right \}}

,

whereby is understood as. ${\ displaystyle \ infty}$ ${\ displaystyle {\ frac {1} {0}}}$

Two nodes and are connected by an edge if and only if ${\ displaystyle {\ frac {a} {c}}}$ ${\ displaystyle {\ frac {b} {d}}}$

{\ displaystyle \ det \ left ({\ begin {array} {cc} a & b \\ c & d \ end {array}} \ right) = \ pm 1}

applies.

Farey sequences are described by Farey diagrams , the Farey graph is the union of all Farey diagrams. ${\ displaystyle F_ {n}}$ ${\ displaystyle \ cup _ {n} F_ {n}}$
In the theory of continued fractions , the Farey graph is used to prove that every periodic continued fraction is a square irrational number .
The modular group and its quotient act by broken-linear transformations , forming adjazente node of the Farey graph back to the node from adjazente. ${\ displaystyle SL (2, \ mathbb {Z})}$ ${\ displaystyle PSL (2, \ mathbb {Z}) = SL (2, \ mathbb {Z}) / \ pm I}$ ${\ displaystyle \ mathbb {Q} \ cup \ left \ {\ infty \ right \}}$

The embedding of the Farey graph in the compactification of the hyperbolic plane by means of the identification and realization of the edges as geodesics gives the Farey tessellation of the hyperbolic plane. ${\ displaystyle \ mathbb {Q} \ cup \ left \ {\ infty \ right \} = P ^ {1} \ mathbb {Q} \ subset P ^ {1} \ mathbb {R} = \ partial _ {\ infty } H ^ {2}}$

The Coxeter group (i.e. the reflection group of an ideal triangle ) shows through on the Farey graph ${\ displaystyle G = \ langle a, b, c \ vert a ^ {2} = b ^ {2} = c ^ {2} = 1 \ rangle}$

{\ displaystyle a \ to \ pm \ left ({\ begin {array} {cc} 0 & -1 \\ 1 & 0 \ end {array}} \ right), b \ to \ pm \ left ({\ begin {array} {cc} 1 & -1 \\ 2 & -1 \ end {array}} \ right), c \ to \ pm \ left ({\ begin {array} {cc} 1 & -2 \\ 1 & -1 \ end {array }} \ right)}

,

each of the triangles of the Farey tessellation is a fundamental domain of the action of on the hyperbolic level.

{\ displaystyle G}

The complex of curves of the dotted torus is the Farey graph, the effect of the mapping class group on the complex of curves is the effect of on the Farey graph. ${\ displaystyle SL (2, \ mathbb {Z})}$