Hatcher-Topology-of-Numbers · Problem 6

Symmetries of the Farey Diagram

Mathematics — Algebraic Topology · linear-fractional-transformations

Problem

(a) Find all the transformations in

LF(\mathbb{Z})

that fix the vertex

{}^1/_0

, that is, take this vertex to itself. (b) Find all the transformations in

LF(\mathbb{Z})

that fix

{}^0/_1

. (c) Determine which of the transformations in (a) and (b) are reflections and describe these reflections. (d) Show that if the transformation

T

fixes

{}^x/_y

then

STS^{-1}

fixes

S({}^x/_y)

. (e) Find all the transformations in

LF(\mathbb{Z})

that fix

{}^1/_1

. Check that

T({}^x/_y) = {}^y/_x

is among the transformations you have found.