We generalize the notions of the orbifold Euler characteristic and of the higher-order orbifold Euler characteristics to spaces with actions of a compact Lie group using integration with respect to the Euler characteristic instead of the summation over finite sets. We show that the equation for the generating series of the kth-order orbifold Euler characteristics of the Cartesian products of the space with the wreath products actions proved by Tamanoi for finite group actions and by Farsi and Seaton for compact Lie group actions with finite isotropy subgroups holds in this case as well.
The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to finding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (for example when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups are presented.
AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).
Rigidity of Furstenberg entropy for semisimple Lie group actions