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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 Other events...
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
Location Maths Lecture Room 1
Contact Michael Giudici <[email protected]>
Start Tue, 15 Sep 2009 13:00
End Tue, 15 Sep 2009 13:45
Submitted by Michael Giudici <[email protected]>
Last Updated Fri, 11 Sep 2009 15:28
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