Differential Calculi in Group Algebras and Group Ends

Graphs generalizing Cayley graphs and arising from various actions of groups on themselves are studied. A relationship between such graphs and subalgebras of operators in a group ring is established, which permits one to obtain a formula for the number of ends of such graphs in terms of the dimensions of appropriate character spaces. Examples of group actions and the corresponding graphs are constructed. In particular, under the action by conjugation, the corresponding algebra turns out to be an algebra of derivations. The proposed construction is generalized to Fox derivatives.