Totally non-free actions of the groups and theory of the characters

In opposite to the case of the finite groups there is no well-developed theory of the characters of the infinite groups. It happened, and partially explained the reason why, the representations which equipped with character, intimately related to the adjoint action of the group on its lattice of its subgroups with an invariant measure, and the character is nothing more that the measure of the set of fixed points of the element of the group under that action.
Although the lattice of the subgroups intencive studed in algerbaic context from 30-th, the question about adjoint invariant measure was in the shadowand only now becomes clear the importance of the problem about the list of such measures for one or another groups, and the construction of the corresponding characters. For infinite symmetric groups and some similar groups the problems was solved several years ago but intrinsic (algebraic and dynamic) explanation of the link is appeared now.