210 IV Continued Fractions and Their Uses
Then
ξn=[an,an+ 1 ,...], − 1 /ηn=[an− 1 ,an− 2 ,...],andξn> 1 ,− 1 <ηn<0. Moreover, ifnis even, thenξandηare properly equivalent
toξnandηnrespectively. If we chooseξso that the sequencea 0 ,a 1 ,a 2 ,...contains
each finite sequence of positive integers (and hence contains it infinitely often), then
the corresponding geodesic inMpasses arbitrarily close to every point ofMand to
every direction at that point.
Some much-studied subgroups of the modular group are thecongruence subgroups
Γ(n), consisting of all linear fractional transformationsz→(az+b)/(cz+d)inΓ
congruent to the identity transformation, i.e.
a≡d≡± 1 , b≡c≡0modn.We may in the same way investigate the geodesics in thequotient spaceH/Γ(n).In
the casen=3 it has been shown by Lehner and Sheingorn (1984) that there is an
interesting connection with theMarkov spectrum.
In Section 2 we defined, for any irrational numberξwith convergentspn/qn,
M(ξ)= lim
n→∞
qn−^1 |qnξ−pn|−^1 ,and we noted thatM(ξ)=M(η)ifξandηare equivalent. It is not difficult to show
that there are uncountably many inequivalentξfor whichM(ξ)=3. However, it was
shown by Markov (1879/80) that there is a sequence of real quadratic irrationalsξ(k)
such thatM(ξ) <3 if and only ifξis equivalent toξ(k)for somek.Ifμk=M(ξ(k)),
thenμ 1 <μ 2 <μ 3 <···andμk→3ask→∞. Althoughμkis irrational,μ^2 kis
rational. The first few values are
μ 1 = 51 /^2 = 2. 236 ..., μ 2 = 81 /^2 = 2. 828 ...,
μ 3 =( 221 )^1 /^2 / 5 = 2. 973 ..., μ 4 =( 1517 )^1 /^2 / 13 = 2. 996 ....As we already showed in Section 2, we can takeξ(^1 )=( 1 +
√
5 )/2andξ(^2 )= 1 +√
2.
Lehner and Sheingorn showed that the simple closed geodesics inH/Γ( 3 )are
just the projections of the geodesics inH whose endpointsξ,ηon the real axis are
conjugate quadratic irrationals equivalent toξ(k)for somek.
There is a recursive procedure for calculating the quantitiesμkandξ(k).AMarkov
tripleis a triple(u,v,w)of positive integers such that
u^2 +v^2 +w^2 = 3 uvw.If(u,v,w)is a Markov triple, then so also are( 3 uw−v,u,w)and( 3 uv−w,u,v).
They are distinct from the original triple ifu=max(u,v,w), since thenu< 3 uw−v
andu< 3 uv−w. They are also distinct from one another ifw<v. Starting from
the trivial triple( 1 , 1 , 1 ), all Markov triples can be obtained by repeated applications
of this process. The successive values ofu=max(u,v,w)are 1, 2 , 5 , 13 , 29 ,....The
numbersμkandξ(k)are the corresponding successive values of( 9 − 4 /u^2 )^1 /^2 and
( 9 − 4 /u^2 )^1 /^2 / 2 + 1 / 2 +v/uw.