Manifolds of Nonpositive Curvature pp 99-102 | Cite as

# 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** ≃ κ_{1}(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).

## Keywords

Compact Manifold Discrete Group Conjugate Point Complete Riemannian Manifold Nonpositive Curvature## Preview

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