Discrete groups of isometries and the Margulis lemma

Abstract

Let V be a complete Riemannian manifold of nonpositive curvature. Let X be the Riemannian universal covering and Γ ≃ κ₁(V) the group of deck transformations. Then Γ is a discrete group of isometries acting freely on X and we can identify V with the quotient X/Γ. On the other hand, if Γ is a discrete group which acts freely on a Hadamard manifold X, then X/Γ is a complete manifold of nonpositive curvature. Let κ: X → V be the canonical projection, let p ∈ V and x ∈ X be a point with κ(x) = p. If c: [a,b] → V is a geodesic loop at p = c(a) = c(b), then let c̃: [a,b] → X be the lift of c with c̃(a) = x. Clearly c̃(b) = γx for an element γ ∈Γ. Thus the geodesic loops at p correspond bijectively to the geodesic segments from x to γx, γ ∈Γ. Because the norm ∣J∣ of Jacobi-fields is convex (§1), there are no conjugate points in V and hence the injectivity radius is half the length of the shortest geodesic loop at p. (Compare [Cheeger-Ebin, 1975] p. 95). Thus, if we define <Inline>1</Inline>, then d _Γ (x) = 2 Inj Rad (κ(x)), where Inj Rad is the injectivity radius on V = X/Γ. A similar discussion shows that the convexity-radius at κ(x) is equal to 1/4 d _Γ (x).