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SUMMARY:Short presentations of finite groups - Alex Lubotzky
DTSTART:20110428T160000Z
DTEND:20110428T173000Z
UID:TALK28061@talks.cam.ac.uk
DESCRIPTION:Finding 'nice & compact' presentations of various groups has b
 een a subject of much research for more than a century. The Coxeter presen
 tation of the symmetric groups and the Steinberg presentation of groups of
  Lie type are such. In response to conjectures of Babai and Szemeredi on o
 ne hand (motivated by questions in computational group theory) and of Mann
  on the other hand (motivated by questions on subgroup growth) we show tha
 t all non-abelian finite simple groups (with the possible exception of Ree
  groups) have presentations which are small (bounded number of relations) 
 and short (w.r.t the length of the relations). This is surprising as the s
 imple abelian groups - the cyclic groups of prime order - do not have such
  presentations! We will describe the motivations and results\, a cohomolog
 ical application (proving a conjecture of Holt) and some connections with 
 discrete subgroups of Lie groups and topology.\n\nThe talk will be suitabl
 e for undergraduates with basic knowledge of group theory.\n\nThe talk is 
 based on a series of papers with Bob Guralnick\, Bill Kantor and Martin Ka
 ssabov.\n\n   1. Presentations of finite simple groups: profinite and coho
 mological approaches\, Groups\, Geometry and Dynamics 1 (2007) 469-523.\n 
   2. Presentations of finite simple groups: a quantitative approach\, J. o
 f the AMS 21 (2008) 711-774.\n   3. Presentations of finite simple groups:
  a computational approach\, J. of the Euro. Math. Soc. 2 (2011) 391- 458.
LOCATION:Wolfson Room (MR 2) Centre for Mathematical Sciences\, Wilberforc
 e Road\, Cambridge
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