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However, Ratner’s Theorem also tells us that s∗ µ is L-invariant, so, because L = H, we conclude that s∗ µ is H-invariant. Thus, H/Λ has an H-invariant probability measure, namely s∗ µ, so Λ is a lattice in H. Thus, Λ has finite index in Λ, so H/Λ is a finite cover of H/Λ = M . 30 4. FUNDAMENTAL GROUPS I Since the action is engaging, and s : M → H/Λ is a G-equivariant section, this implies H/Λ = M , so Λ = Λ. M. Gromov provided another important class of topologically engaging examples. We will see the proof in Lecture 5.

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References  D. Fisher: On the arithmetic structure of lattice actions on compact spaces, Ergodic Theory Dynam. Systems 22 (2002), no. 4, 1141–1168. MR 1926279 (2004j:37004) REFERENCES 51  D. Fisher and K. Whyte: Continuous quotients for lattice actions on compact spaces, Geom. Dedicata 87 (2001), no. 1-3, 181–189. MR 1866848 (2002j:57070)  A. Lubotzky and R. J. Zimmer: A canonical arithmetic quotient for simple Lie group actions, in: S. G. , Lie Groups and Ergodic Theory (Mumbai, 1996).