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We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affne twin buildings of rank at least 3., Comment: 20 pages, preliminary version, to appear in "Advances in Geometry"

A twin building consists of two buildings that are twinned by a codistance function. We prove that the local structure of a twin building uniquely determines the two buildings up to isomorphism. This has been known for twin buildings satisfying a technical condition (co)., Comment: 14 pages

We investigate the geometry in a real Euclidean building X of type A2 of some simple configurations in the associated projective plane at infinity P, seen as ideal configurations in X, and relate it with the projective invariants (from the cross ratio on P). In particular we establish a geometric classification of generic triples of ideal chambers of X and relate it with the triple ratio of triples of flags., Comment: to appear in Innovations in Incidence Geometry

If a geometry $\Gamma$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $\Delta$, a representation $\varepsilon_\Delta^A$ of $\Gamma$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(\Delta)$ of a flag $A^*$ of $\Delta$ opposite to $A$, every element of $\Gamma$ being mapped onto a suitable subgroup of $U_{A^*}$. We call such a representation a unipotent representation. We develope some theory for unipotent representations and we examine a number of interesting cases, where a projective embedding of a Lie incidence geometry $\Gamma$ can be obtained as a quotient of a suitable unipotent representation $\varepsilon_\Delta^A$ by factorizing over the derived subgroup of $U_{A^*}$, while $\varepsilon^A_\Delta$ itself is not a proper quotient of any other representation of $\Gamma$., Comment: 35 pages

Based on the second author's thesis 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 by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov.

This book is dedicated to Grassmannians associated with buildings of classical types: usual, polar, and half-spin Grassmannians. Grassmannians of vector spaces and Grassmannians consisting of totally isotropic subspaces of non-degenerate alternating, Hermitian, and symmetric forms are special cases of these "building" Grassmannians., Comment: This paper has been withdrawn by the author. 117+iii pages, 5 figures Full version(more than 200 pages) was published by World Scientific

We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups.

For a local field $K$ and $n \geq 2$, let $\Xi_n$ and $\Delta_n$ denote the affine buildings naturally associated to the special linear and symplectic groups $\SL_n(K)$ and $\Sp_n(K)$, respectively. We relate the number of vertices in $\Xi_n$ ($n \geq 3$) close (i.e., gallery distance 1) to a given vertex in $\Xi_n$ to the number of chambers in $\Xi_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in $\Delta_n$ ($n \geq 2$) close to a given special vertex in $\Delta_n$ (all the vertices in $\Xi_n$ are special) and establish analogues of our results for $\Delta_n$., Comment: 16 pages, 3 figures; minor corrections; accepted for publication in INTEGERS: The Electronic Journal of Combinatorial Number Theory

This is a survey article on classical groups (over arbitrary division rings) and their geometries.