# Extensions of diagram geometries and existence proofs for finite simple groups

## Description of the research project

The aim of this project is to study structures of parabolic systems related to diagram geometries.

1) Classification of the semiclassical parabolic systems. These are parabolic systems {P_1,....,P_n} such that B = P_1 \cap ..... \cap P_n = P_i \cap P_j , P_i/B_{P_i} are rank one Lie groups in characteristic 2, for all i,j, /B_{\langle P_i,P_j\rangle} is a rank two Lie group in characteristic 2 or 3S_6 and $B/B_{\langle P_i,P_j \rangle}$ is the Borel subgroup.

Such a system is called semiclassical if the case /B_{\langle P_i,P_j \rangle} = 3S_6 really occurs.

Interesting groups occurring in this context are : M_24, He, Co_1, F_1, M(24).

2) c-extensions of P-Geometries : We consider Petersen geometries in the sense of Ivanov, Shpectorov. These are geometries for M_22, 3M_22, M_23, J_4, Co_2, 3^23Co_2, F_2 , 3^4471F_2. For all these geometries c-extensions are considered. We like to find examples and if possible to classify them. For example M_22 yields the universal representation group (2^11M_22(2)) but also M_24 and 2U_6(2)(2). M_23 yields M_24 (besides the representation group). Co_2 yields the representation group (2^23Co_2) and Co_1 and F_2 yields the representation group (2(F_2 x F_2)2) and F_1.

3) Construction of groups : Using the geometries above and maybe some others we try to construct the sporadic simple groups in an uniform way. Here first of all we look for a representation which somehow is related to the geometry and then embed the amalgam into the corresponding GL(n,K). The main problem afterwards is to prove that the embedded amalgam really is the corresponding sporadic group. This for example has been done for J_1(Z. Janko), J_3(B. Baumeister), J_4(U. Meierfrankenfeld, A. Ivanov). There has been work done for J_2, Suz and Co_1. The group O'N is under investigation. Next groups to be considered He, M(22), M(23), M(24).

## This is joint work partly done together with

B. Baumeister (Imperial College, London)

A. Fukshansky (Halle) (DFG)

A. Pasini (Siena)

S. Shpectorov (Bowling Green, OH, USA)

C. Wiedorn (Halle) (DFG)

The following preprint related to the project may be viewed by clicking below:

The noncanonical gluings of affine spaces (Joint with Barbara Baumeister)

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A characterization of 3-local geometry of $M(24)$ (Joint with A.A. Ivanov)

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Semiclassical parabolic systems related to $M_{24}$ ( Joint with Anna Fukshansky)

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$c$--extensions of $P$--geometries (Joint with Corinna Wiedorn)

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$c$ - extensions of the Petersen geometry for $M_{22}$(Anna Fukshansky, Corinna Wiedorn)

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The semiclassical parabolic systems (Joint with Anna Fukshansky and Corinna Wiedorn)

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Affine dual polar spaces (Joint with Barbara Baumeister and Sergey Shpectorov)

Latex File in preparation