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GABAI'S ONE-SIDED LAMINATION It is well known that a properly embedded line in the plane separates the plane into two regions.
 

Summary: GABAI'S ONE-SIDED LAMINATION
It is well known that a properly embedded line in the plane separates the plane into two regions.
A measured lamination is an object which was introduced by Thurston which looks locally like
a Cantor set I. Gabai found a measured lamination in the plane whose complement is connected.
To construct a measured lamination, take a train track with weights on each branch, such that the
sum of incoming weights equals the outgoing weights at each exchange. Then thicken up the train
track so that the width of the cross-section of each branch is equal to its weight, and finally split so
that one gets the lamination (see figure 1).
Figure 2 shows an infinite measured train track in the plane such that the associated lamination
is one-sided. After the splitting and isotopy indicated in figure 2, one obtains the same train track
with the weights multiplied by 1
2 , and such that the top tear drop is now connected to the unbounded
component. Continuing like a conveyor belt, one can transfer any teardrop complementary region
1
1/2
1/2
1
1/2
1/2

  

Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

 

Collections: Mathematics