In his notes ``On the geometry and dynamics of diffeomorphisms of surfaces'',
Thurston describes a method for constructing surfaces (generally
half-translation surfaces) for whom a large subgroup of the Veech group,
isomorphic to the free group on two generators, is known.
(See also:
BouillabaisseSurfaces.)
Let
and
be transverse (not necessarily maximal) multicurves
on a compact orientable surface
, and assume that they
fill the surface,
i.e., that each component of their complement
is a
simply connected polygon. Because
and
are transverse to each
other, the edges of the cylinders will also be transverse to each other, and will
cut each other into rectangles. The corners of these rectangles meet inside the
polygons that form the complement of
and
, and so the number
of sides of the polygon determines what kind of singularity it will contain.
We want to find appropriate side lengths of these rectangles.
Suppose
and
, and let
![$M = [m_{ij}]$](/~twiki/pub/WildSurfaces/ThurstonSurface/latexe16caf2d2cf320341da6e124adce4f5e.png)
be the
matrix such
that
is the number of crossings of
and
. Thus
counts the number of rectangles comprising
that contain a
crossing of
and
. Now consider the matrix
;
it is symmetric, which means all of its eigenvalues are real. We let
denote the largest eigenvalue of
. Choose an eigenvector
of
corresponding to
and having positive entries, and set
. The calculation
shows that
is an eigenvector of
, also with eigenvalue
.
Now let each rectangle that contains an intersection of
and
have width
and height
. (We now assume that
is horizontal and
is vertical with respect to the flat structure.)
The following proposition shows that these coordinates are the correct ones.
Proposition.
With respect to this flat structure, the Dehn twists around
and
are affine automorphisms, and their derivatives have the forms
These two Dehn twists are parabolic transformations. The group they generate
contains other parabolic elements, which are conjugates of the generators; all
other elements are hyperbolic (pseudo-Anosov).
Variants of this construction are possible. For example, each curve can
be assigned a positive integral
weight, so that the corresponding parabolic
transformation consists of different multiples of a Dehn twist for each curve.
One can also choose
for some positive real number
; then
, where
. But this just amounts to applying
, for some real
, to the surface described
above.
--
JoshuaBowman - 17 Feb 2006
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![\[ MM^\top w = MM^\top Mv = M(\lambda v) = \lambda Mv = \lambda w \]](/~twiki/pub/WildSurfaces/ThurstonSurface/latex1ef61922edd84e95f046eb44584ecc53.png)
![\[ D_\Gamma = \left( \begin{array}{ c c } 1 & \lambda \\ 0 & 1 \end{array} \right) \ \ \mbox{and}\ \ D_\Delta = \left( \begin{array}{ c c } 1 & 0 \\ -1 & 1 \end{array} \right). \]](/~twiki/pub/WildSurfaces/ThurstonSurface/latex66911439bbb85371128d639af5243bcf.png)