Hyperbolic lines in long root group geometries
by
Ralf Gramlich
TU Eindhoven, The Netherlands

Let n >= 11, let F be a skewfield and let (P, L, \perp ) be a partial linear space endowed with a symmetric relation \perp on the point set P such that for x a point and l a line x \perp p in l and x \perp q in l \{ p } implies x \perp y for all y in l. If for any line k in L the space k ^\perp is isomorphic to the hyperbolic root group geometry of PE_n+1(F) with l \perp m if and only if [ l, m ] = 1 for lines l, m inside k ^\perp , and the graph (L, \perp ) is connected, then (P, L) is isomorphic to the hyperbolic root group geometry of PE_n+3(F).

Posing additional conditions, I can decrease n. The ideas used in the proof can be adjusted to characterize the hyperbolic long root group geometries of any classical group, for high enough dimension. There might even be hope for a characterization of the geometries of some exceptional groups.

Date received: January 10, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-02.