Bruhat-Tits spaces (Q1124694)

The parallelogram rule in the Euclidean plane is the statement that the sum of the squared lengths of the two diagonals of a parallelogram equals the sum of the squared lengths of the four sides. This condition can be weakend to an inequality which makes sense in any metric space \(X\): We say that a metric space \(X\) with distance function \(d\) satisfies the semi-parallelogram rule if for any two points \(x,y\) (in \(X\)) there exists a point \(z\) such that for all \(w\) the quantity \(d(x,y)^2 + 4d(w,z)^2\) is less or equal to \(2d(w,x)^2 + 2d(w,y)^2.\) A complete metric space with this property is called a Bruhat-Tits space. This expository paper is centred around the fixed point theorem of Bruhat-Tits, which affirms that a group of isometries of a Bruhat-Tits space with a compact orbit has a fixed point. First an elementary proof of that theorem is given. Then the question of existence of non-Euclidean Bruhat-Tits spaces is addressed. It is known that any complete simply connected Riemannian manifold of non-positive (semi-negative in the terminology of the paper) sectional curvature is a Bruhat-Tits space. (In that case the above fixed point theorem is known as Cartan's fixed point theorem.) An example of such a manifold is the symmetric space of positive definite real symmetric matrices. It is discussed in detail without assuming any knowledge of Riemannian geometry. Therefore the text is well suited for students who have just learned Infinitesimal Calculus and Linear Algebra . It may provoke their curiosity towards the wonderful world of geometry and Lie groups.