Graphs connected to the hadamard manifold and groups of its isometries

Abstract

Three graphs are associated to the Hadamard manifold X and the discrete group of its isometries Γ.The set of vertices of the first graph is X itself, of the second all the almost nilpotent subgroups in Γ,and of the third the geometric invariants of the vertices of the second graph such as the fixed point set, collections of invariant lines, and so on. The group Γacts on all three graphs and there exist images of these graphs that are equivariant with resepct to these actions. This formalism allows us to adduce a simple proof of the following theorem. Let M be an n-dimensional complete Riemannian manifold with sectional curvatures 1 ≤ K ≤ 0 and satisfying the visibility axiom. Then there exists a point p ∈M such that the injectivity radius Inj Rad(p) ≥ c(n), where the constant c(n) > 0 depends only on n. Other results obtained with the help of this formalism are also given.