Surfaces of large genus
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Mon, 31/01/2011 15:45 |
Hugo Parlier (University of Fribourg) |
Topology Seminar  |
L3 |
Surfaces of large genus are intriguing objects. Their geometry
has been studied by finding geometric properties that hold for all
surfaces of the same genus, and by finding families of surfaces with
unexpected or extreme geometric behavior. A classical example of this is
the size of systoles where on the one hand Gromov showed that there exists
a universal constant  such that any (orientable) surface of genus 
with area normalized to has a homotopically non-trivial loop (a
systole) of length less than  . On the other hand, Buser and
Sarnak constructed a family of hyperbolic surfaces where the systole
roughly grows like  . Another important example, in particular for
the study of hyperbolic surfaces and the related study of Teichmüller
spaces, is the study of short pants decompositions, first studied by Bers.
The talk will discuss two ideas on how to further the understanding of
surfaces of large genus. The first part will be about joint results with
F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of
pants decompositions and related questions. In particular we investigate
how to find short pants decompositions on punctured spheres, and how to
find families of homologically independent short curves. The second part,
joint with L. Guth and R. Young, will be about how to construct surfaces
with large pants decompositions using random constructions. |
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