SEMINAR: Groups and Combinatorics Seminar: Tits' buildings and applications to the classification of finite simple groups
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Groups and Combinatorics Seminar: Tits' buildings and applications to the classification of finite simple groups |
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Groups and Combinatorics Seminar
Alice Devillers
will speak on
Tits' buildings and applications to the classification of finite simple
groups
at 1pm on Tuesday 15th of September in MLR1.
Abstract:
For a field K, the group PSL(n+1,K) acts on the projective space
PG(n,K). For K=C or R, that group is a simple Lie group. Jacques Tits,
Belgian mathematician, invented a combinatorial structure, generalising
projective spaces, on which the semisimple Lie groups act naturally.
These combinatorial objects can be seen as simplicial complexes or as
chamber systems and are called buildings.
Tits' Lemma states that for a group acting transitively on a chamber
system, the chamber system is simply connected if and only if the group
can be `nicely presented' as an amalgam.
Since buildings are simply connected, this provides presentations of
semisimple Lie groups. There are other groups acting transitively on
some subchamber systems of buildings.
With Bernhard Muhlherr, we provided a local combinatorial criterion for
a chamber system to be simply connected. This criterion is then used by
Bennett, Gramlich, Hoffman, Shpectorov to get nice presentations of
these groups, called Curtis-Tits-Phan presentations.
Most finite simple groups (in particular the classical ones) have an
action on buildings or on subchamber systems of buildings. Hence we get
nice amalgam presentations for them.
This allows a `local recognition' of these groups and is used in the
ongoing reproving of the classification of finite simple groups
(Gorenstein, Lyons and Solomon).
This talk will be an extended version of the talk I will give in
Adelaide for AustMS. It is also related to the subject of our new
Buildings, finite geometries and Phan-systems brain-storming sessions.
All welcome
Speaker(s) |
Alice Devillers
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Location |
Maths Lecture Room 1
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Contact |
Michael Giudici
<[email protected]>
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Start |
Tue, 15 Sep 2009 13:00
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End |
Tue, 15 Sep 2009 13:45
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Submitted by |
Michael Giudici <[email protected]>
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Last Updated |
Fri, 11 Sep 2009 15:28
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