Index abelian group, 36, 80, 81 abuse of terminology, 26, 67 admissible measure, 61, 62 aﬃnely flat manifold, 53, 54 algebraic action, 19, 78, 79 group, 17, 18, 33, 41, 43, 77–79 hull, 19–22, 33–37, 39, 46 Cr, 19, 37 kth order, 37 almost-invariant vector, 28, 81 amenable equivalence relation, 72 group, 27–28, 30, 31, 45, 62, 68, 70, 72, 73, 80–82 analytic continuation, 38 function, see real analytic set, 12, 13 Anosov flow, 26 arithmetic action, 50 group, 1, 10, 49, 50, 65 atomic measure, 19 Auslander Conjecture, 54, 58 automorphism infinitesimal, 38 local, 38 of C, 2, 43, 79, 80 ax + b group, 21 Banach space, 80 big lattice, 11–13 blowing up, 4–5, 30 Borel subalgebra, 75 bounded cohomology, 11, 14, 72, 73 Cantor set, 12 Cartan geometry, 22 Cartan matrix of an inner product, 18 Cayley graph, 71 centralizer, 11, 12, 33–35, 37–39, 42, 55 circle S1, 10–14 bundle, 63 cocycle, 41, 45, 68, 78 cohomologous, 41, 43, 45, 68, 78 identity, 41 cohomology group Hk(M R), 21 commensurable groups, 47, 49, 71, 72, 79, 80 compact group, 2, 27, 33, 35, 42–44, 47, 48, 54, 56, 61–63, 69, 76, 79, 81 lattice form, 54–58 complete form, see compact lattice form connection (on a principal bundle), 30, 33–38, 43–46 rigidity of, 34, 38 convex set, 62, 80, 81 convolution of measures, 61, 62 cost of an equivalence relation, 70, 72 covering space, 25, 29, 30, 49 double, 30 finite, see finite cover regular, 27 universal, 25–38, 47, 48, 71, 75, 80 descending chain condition, 19, 26 engaging, 25–30, 47, 48, 50, 65 geometrically, 38 topologically, 26–27, 29–30, 33, 35, 38, 47, 50 totally, 26, 29–30, 47, 49, 50 entropy finite, 50 Furstenberg, see Furstenberg entropy equivalence relation, 67, 69, 70, 78 amenable, see amenable equivalence relation ergodic, definition of, 76 Euler characteristic, 12–14 finite action, 9–14, 43, 44 cover, 26, 27, 29, 43, 47 orbit, 9–13 quotient, 26 85

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