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RESEARCH BLOG 11/16/03 This weekend, Howie Masur, Benson Farb, and Alex Eskin hosted
 

Summary: RESEARCH BLOG 11/16/03
This weekend, Howie Masur, Benson Farb, and Alex Eskin hosted
a conference on Teichmuller theory here at UIC. There were many
interesting talks. Probably the most spectacular result was by Ur-
sula Hamenstaedt , who claims to have proven that the mapping class
groups of surfaces S are rigid. This was proven by Mosher and Whyte
in some special cases. Her strategy is to use the complex of train
tracks, which are train tracks in a surface which are recurrent and are
not sub-train train tracks of another (in a non-trivial way), with edges
of the complex given by splittings (see blog 2/27/03). She shows
that MCG(S) acts faithfully cocompactly on this train track com-
plex, and she constructs a bi-combing which satisfies some other nice
properties (that I didn't quite catch), which are satisfied by groups
which act faithfully co-compactly on CAT(0) spaces (although map-
ping class groups of surfaces of genus > 2 cannot be CAT(0), by a
result of Geoff Mess). From this, she can construct the asymptotic
cone, and its boundary. This result was also announced in a talk by
Jason Behrstock last Monday here at UIC, using different techniques
(Note added 11/21/03: he computes the asymptotic cone and the max-
imal rank of flats, but doesn't get QI rigidity). The asymptotic cone

  

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

 

Collections: Mathematics