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We deduce from Sageev’s results that whenever a group acts locally elliptically on a finite-dimensional CAT cube complex, then it must fix a point. As an application, we partially prove a conjecture by Marquis concerning actions on buildings and we give an example of a group such that does not have property (T), but and all its finitely generated subgroups can not act without a fixed point on a finite-dimensional CAT cube complex, answering a question by Barnhill and Chatterji.
We classify affine rank three Tits arrangements whose roots are contained in the locus of a homogeneous cubic polynomial. We find that there exist irreducible affine Tits arrangements which are not locally spherical.
We explore the minimal characteristic two parabolic geometries for the finite sporadic simple groups, as introduced by Ronan and Stroth. The chamber graphs of the geometries are studied, with the aid of Magma, focusing on their disc structure and geodesic closures. For the larger sporadic geometries which are beyond computational reach we give bounds on the diameter of their chamber graphs.
For we determine the independence number of the Kneser graph on plane-solid flags in PG. More precisely we describe all maximal independent sets of size at least and show that every other maximal example has cardinality at most a constant times .