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The aim of this e-book is to illustrate that advanced numbers and geometry may be mixed jointly fantastically. This leads to effortless proofs and common generalizations of many theorems in aircraft geometry, comparable to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.
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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).
A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H.