DFG Research Projekts

Transversals in Groups with an application to loops

GZ BA 2200/2-2, January 2007 - January 2010
A Post Doc position: Dr. Alexander Stein

Abstract: Groups with a regular subset have been studied for a long time, in particular groups related to finite Bruck loops, see [A05], [GG1}. A loop can be thought of as a group without associativity. These are very wild structures. A Bruck loop satisfies a weak associativity and the so called automorphic inverse property. These structures appear at many places in mathematics as well as in physics. That is probably why they had been invented several times! Within this DFG-project Stein and myself could accomplish the classification of the groups related to finite Bruck loops [BS10b], [BS10a], [BSS11] which was started by Glauberman [G64, G68] and Aschbacher [A05].

[A05] M. Aschbacher, On Bol loops of exponent 2, J. Algebra 288 (2005), 99 - 136
[BS11] B. Baumeister, A. Stein, The finite Bruck loops, J. Algebra 330 (2011), 206 - 220
[BS10a] B. Baumeister, A. Stein, Commuting graphs of odd order elements in simple groups, arXiv: 0908.2583, Januar 2010, 26 Seiten, eingereicht
[BS10b] B. Baumeister, A. Stein, Self-invariant 1-Factorizations of complete graphs and finite Bol Loops of Exponent 2, Beitraege zur Algebra und Geometrie 51 (2010), 117 - 135
[BSS11] B. Baumeister, A. Stein, G. Stroth, On Bruck loops of 2-power-exponent, J. Algebra 327 (2011), 316 - 336
[G68] G. Glauberman, On Loops of odd order II, J. Algebra 8 (1968), 393 - 414
[G64] G. Glauberman, On loops of odd order, J. Algebra 1 (1964), 374 - 396

Moufang Sets, split BN-pairs of rank one, finite and infinite doubly transitive groups

GZ BA 2200//3-1, January 2010 -
A Post Doc position: Dr. Matthias Grueninger

Abstract: Mark Ronan und Jaques Tits [RT94] introduced the notation of a twin tree. They are natural geometric objects to describe Kac-Moody groups and algebraic groups and algebraic groups over the ring of (skew) Laurent polynomials [Ti92]. My Post Doc and myself study the geometric relation between these notions [BGM11]. The ends of a Moufang twin tree are quite often Moufang sets. Moufang sets can be seen as rank one Moufang buildings or as doubly transitive permutation groups such that the point stabilizer contains a regular normal subgroup, i.e. as finite doubly transitive groups with a split BN-pair of rank one. We also use this new theory to simplify the known classification of the doubly transitive permutation groups [BG11].

[BG11] B. Baumeister, M. Grueninger, Finite Zassenhaus Moufang Sets with root groups of even order, Commun. Contempt. Math., appears in 2011, 30 pages.
[BGM11] B. Baumeister, M. Grueninger, B. Muehlherr, Moufang twin trees and right angled buildings, Draft 20 pages.
[RT94] M. Ronan, J. Tits, Twin Trees I, Invent. Math. 116, No. 1-3, 463-479 (1994).
[Ti92] J. Tits, Twin buildings and groups of Kac-Moody type, in Groups, combinatorics and geometry (Durham 1990), Lond. Math. Soc. Lect. Note Ser. 165 (1992), 249 - 286

The Topology and Geometrie of Groups, SFB 701, Universitaet Bielefeld

Principal Investigators: Barbara Baumeister, Kai-Uwe Bux