Building

Description

Discussion given by Dorian Chanfi

Groups are historically used to study a Coxeter system. These are a symplicial complex that capture some property of the group. We are given a presentation.

Example

As an example we study the Symmetriegruppe $S_3$. Its coxeter complex looks like a hexagon. Given a dihedral group its coxeter complex looks like a line.

These Coxeter systems will be the slices/ingredients of which our buildings will be made.

Definition

A building is a simplicial complex $\Delta$ and a family of subcomplexes (apartments) s.t.

1. Each Apartment is isomorphic to a Coxeter system
2. For any two maximal simplices (chamber), there is an apartment contained in both.
3. Given two apartments, there is a isomorphism fixing the union apartment pointwise.

A building is called spherical if it looks like a sphere.

Example

We take an example from spherical geometry.

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Theorem (Gruhat-Tits)

If $G$ is a semisimple group over a $k$ non—archimedean local field then $G(k)$ acts properly, strongly transitively on an affine building.

This theorem states, that under the correct circumstances buildings are quite natural indeed.

Facts:

• Affine buildings have a natural CAT(0) actions

Properties

Tip

Examples

Example