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### Describing the action of $^2E_6(q)$

One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
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### Buildings for Affine groups

Let $G$ denote one of the classical groups over a finite field. Is there a natural way to associate a building to the affine group $V\rtimes G$, and an analog of the Solomon-Tits theorem?
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### Pairs of rays in euclidean buildings

In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...
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### Is a cocompact CAT(0) periodic?

Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...
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### Understanding how to construct Bruhat-Tits buildings for non-split groups by Galois descent

Is there any way to get on top of the procedure for constructing Bruhat-Tits buildings for non-split groups over a non-archimedean local field $k$, by Galois descent, other than reading both the ...
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### Algorithm for the cell multiplication rule for GL(n,F)

Consider $F$ a non archimedean field and let $o$ be its ring of integer Let $B$ be the Iwahori subgroup of $GL_n(F)$ (resp. $GL_n(o)$) and let $N$ be the normalizer of the diagonal matrices (...
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### What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?

$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$? This can be phrased also as question about ...
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### Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...
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Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of $\... 6answers 4k views ### Any good reference for Tits Building? For beginers, any suggestions? 3answers 360 views ### Is this the CAT(0) metric on an affine building? Let$R$be a discrete valuation ring qith quotient field$Q$and let$t\in R$be a generator of the unique maximal ideal in$R$. Let$V$be a finite-dimensional$Q$-vector space. Then one can consider ... 1answer 296 views ### On which space does$GL_n(F_p[X])$act nicely? The group$GL_n(\mathbb{Z})$acts properly and isometrically on the space of homothety classes of scalar products on$\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature. ... 1answer 782 views ### Proof of an 'easy' exercise in a book of Tits In 'Buildings and Finite$BN$-Pairs', Jacques Tits gives the following statement which is left as an easy exercise. Let$G_1,G_2,G_3$be three subgroups of a group$G$. Then the following conditions ... 2answers 699 views ### Geometric interpretation of$BN$-pairs My question is relative to a geometric interpretation of the$BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).$[...
Here $F$ is a locally compact non-archimedean non-discrete field. Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...