On the simple connectedness of certain subsets of buildings

Simple connectivity in polar spaces

“…With some effort, and using a straight-forward generalization of [5], it should be possible to prove that the geometry induced by the set of chambers opposite a given chamber in any polar space of rank at least 4 is simply connected, with the sole exception of W (7, 2).…”

“…Our results imply that this condition is satisfied for the building corresponding to the thick non-embeddable polar spaces. Similarly, but using different methods, Devillers & Mühlherr [5] prove that, the (positive or negative) Borel subgroup of every Kac-Moody group defined over a finite field with at least 16 elements is finitely presented. This bound can now be sharpened to "at least 5 elements".…”

Abstract involutions of algebraic groups and of Kac–Moody groups

“…Another important class of quasi-flips has first appeared in [24]: Definition 3.13 ([24, Definition 6.2]). Let y be a building quasi-flip of a twin building C. For any spherical residue R, define proj R ðyÞ :¼ fc A R j proj R ðyðcÞÞ ¼ cg, where proj R denotes the projection onto R. If for all panels P of C one has proj P ðyÞ 0 P, one calls y a strong quasi-flip.…”

“…Based on the second author's thesis [33], in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [32] and [49] for algebraic groups and in [39] for certain Kac-Moody groups, we analyze the filtration studied in [24] in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised in [7]. …”

The sphericity of the Phan geometries of type $B_n$ and $C_n$ and the Phan-type theorem of type $F_4$

“…We use a filtration (∆ i ) i of ∆ + with ∆ 0 = ∆ θ from [DM07], whose relative links will turn out (Theorem 5.10) to be generalized Phan geometries (cf. [DGM09,Fact 5.1]).…”