# Discrete groups of isometries and the Margulis lemma

• Werner Ballmann
• Mikhael Gromov
• Viktor Schroeder
Part of the Progress in Mathematics book series (PM, volume 61)

## 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).

## Keywords

Compact Manifold Discrete Group Conjugate Point Complete Riemannian Manifold Nonpositive Curvature
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Werner Ballmann
• 1
• 2
• Mikhael Gromov
• 3
• Viktor Schroeder
• 4
• 5
1. 1.Dept. of MathematicsUniversity of MarylandCollege ParkUSA
2. 2.Math. Institut der UniversitätBonnWest Germany
3. 3.Inst. des Hautes Etudes ScientifiquesBures-sur-YvetteFrance
4. 4.Math. Institut der UniversitätMünsterGermany
5. 5.Math. Institut der UniversitätBaselSwitzerland