The explicit computations of the modular obstruction for G-kernel on a certain AFD factor

设N为群G的正规子群，其商群Q=G/N。在以前的工作中，用连续交叉积给出了模阻塞的定义。根据a . connes对AFD iii_λ -因子的离散分解，我们可以考虑群Q_m={(p,t)∈Q×R: mod α_p=t mod t 'Z}。因此G上的扩展模阻塞与Q_m上的某环面值为3上同构，G上的特征不变量也是如此。然后，我们得到了III_λ型afd因子的扩展hgr精确序列。这个精确序列是下面图中的第一列，显示了它与普通hgr -精确序列的关系：【数值公式】我们通过给出群类群的归纳约简理论及其维数变换定理证明了上面的图，另一方面，这个环面值扩展模不变量是一个完全不变量，直到外共轭。对于III_λ型afd因子的外部作用，该离散扩展hj -精确序列帮助我们给出给定模阻的外部作用模型。设A是SL（2z）中的一个元素。我们考虑一个交叉积群G(a)=Z^2×_A Z by a。我们将3-上同调群H^3(G(a)，T) = T参数化。我们明确给出了H^3 （G(A), t）中与t相关的条件下的模型外作用，并给出了如何从给定的外作用中得到元素t的方法。通过构造具有给定不变量的模型作用，我们需要显式地计算G(a)群的hgr -精确序列，例如G(G)群的2-上同群特征不变量。

Let N be a normal subgroup of a group G and its quotient group Q=G/N. In the former work, the definition of modular obstruction was given by the use of continuous crossed product. According to a discrete decomposition of AFD III_λ-factor (by A.Connes), we may consider the group Q_m={(p,t)∈Q×R : mod α_p=t mod T'Z}. Consequently the extended modular obstruction on G is isomorphic to a certain Torus valued 3-cohomology on Q_m and the characteristic invariant on G is also so. Then we have the extended HJR-exact sequence for AFD-factor of type III_λ. This exact sequence is the first column in the following diagram which shows the relation between it and an ordinary HJR-exact sequence :【numerical fomula】We proved the above diagram, by giving induction and reduction theory for groupoids and their dimension shifting theorem, On the other hand, this Torus valued extended modular invariant is a complete invariant, up to outer conjugacy, for outer actions on AFD-factor of type III_λ and this discrete extended HJR-exact sequence help us to give a model of outer action with a given modular obstruction.Let A be an element of SL(2 Z). We consider a crossed product group G(A)=Z^2×_A Z by A. We parameterize the 3-cohomology group H^3(G(A),T)〓T. We give explicitly the model outer action with some condition associated with t in H^3 (G(A),T) and we also give the way how to get the element t from a given outer action. By cotistructing a model action with a given invariant, we have to compute HJR-exact sequence for the group G(A) explicitly, for example, 2-cohomology group characteristic invariant for the group G(G).