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D. Cooper and D.D. Long \Lambda In [1], we described an algorithm to determine whether a certain kind of surface in a 3manifold
 

Summary: Trimming.
D. Cooper and D.D. Long \Lambda
In [1], we described an algorithm to determine whether a certain kind of surface in a 3­manifold
is quasi­fuchsian or a virtual fibre. The surface in question is required to be transverse to the
suspension flow in a closed hyperbolic 3­manifold which fibres, but need not be embedded. One of
the reasons that this question is interesting is that it is connected to showing that hyperbolic surface
bundles have infinite virtual Betti number.
Although the algorithm of [1] is practical in the sense that one can implement it on a variety
of examples, such implementations were rather ad hoc. In this paper we improve the algorithm
somewhat so that it may be more uniformly implemented on a computer. As a consequence we
obtain a method which is sufficiently efficient and practical that one can deal with a broader range
of examples.
The first step of [1] is to reduce the question to one about similarity interval exchange (SIE)
maps; we recall what is relevant of the definition below. To expedite the algorithm, we introduce
a concept called trimming. This produces from a given geometric SIE, a sequence of SIE's which
has the property that after a finite number of trims, either the SIE has a fixed point, (from which
it follows that the surface is quasi­Fuchsian) or else the sequence is eventually periodic (and the
surface is a virtual fibre).
We conclude with a collection of examples to which this algorithm was applied; the results suggest
that ``most'' simple immersed surfaces are quasi­fuchsian. However, any general understanding which

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics