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Receding polar regions of a spherical building and the center conjecture

Bernhard Mühlherr, Richard M. Weiss (2013)

Annales de l’institut Fourier

We introduce the notion of a polar region of a spherical building and use some simple observations about polar regions to give elementary proofs of various fundamental properties of root groups. We combine some of these observations with results of Timmesfeld, Balser and Lytchak to give a new proof of the center conjecture for convex chamber subcomplexes of thick spherical buildings.

Reflection loops of spaces with congruence and hyperbolic incidence structure

Alexander Kreuzer (2004)

Commentationes Mathematicae Universitatis Carolinae

In an absolute space ( P , 𝔏 , , α ) with congruence there are line reflections and point reflections. With the help of point reflections one can define in a natural way an addition + of points which is only associative if the product of three point reflection is a point reflection again. In general, for example for the case that ( P , 𝔏 , α ) is a linear space with hyperbolic incidence structure, the addition is not associative. ( P , + ) is a K-loop or a Bruck loop.

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