CHAPTER 0 Preliminaries OA. Actions Consider an action (7, x) G Y x X 1— • 7 • x of a group V on a set X, sometimes referred to as a V-action or a V-space. It induces an equivalence relation on X, whose equivalence classes are the orbits of the action. The T-saturation of A C X is the set V • A = {7 • x : rr G A, 7 G T}. If T • A = A, we call A T-invariant. The stabilizer of a point x G l , Stab(x) or Tx, is the subgroup of T defined by Tx = Stab(x) = {7 G T : 7 • x — x}. If Stab(x) = {1} for all x G X, i.e., 7 • x 7^ x, V7 G T,x G X, we call the action free. In general the free part of the action of T on X is the T-invariant set {x G X : Stab(z) = {1}}). If T acts on spaces X, Y, a map p : X — Y is called a T-map if ^(7 • x) = 7- /?(», V7 G T,x G X. For any group T and any set X, we have the shift action of X on X r , defined by (7 • f)(S) = /(7~ 1 £). The corresponding equivalence relation is denoted by #(r,x). We let (X) r be the free part of this action, i.e., the set W r = { / e i r : 7 ' / / / , V 7 e r , 7 / i ) . The equivalence relation induced by the action of T on (X) r is denoted by F(T,X). If T acts on each space X^i G 7, the diagonal action of T on Yii X^ is given by: 7 ' (xi) = (l-Xi). If each I\ acts on X^, i G 7, then the product action of [^ I\ on J]^ X^ is given by: (li) ' fa) = (pti - ^ ) - OB. Equivalence relations If E C X 2 is an equivalence relation on a set X, we write interchangeably xEy or (x, i/) G E to indicate that x is equivalent to y. We denote by X/E the quotient space, i.e., the set of its equivalence classes. We also denote by [X]E the equivalence class or E-class of x G X. More generally if A C X, we let [A]E = {xeX:3ye A(xEy)}, and call [A]E the E-saturation of A. If [A] E = A, we call A E-invariant. If [A]£ = X, we call ^4 a complete section of E. Finally, E\A — En A2 is the restriction 9

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