Groups Acting on Cantor Sets and the End Structure of Graphs

We describe a variation of the Bergman norm for the algebra of cuts of a connected graph admitting a cofinite group action. By a construction of Dunwoody, this enables us to obtain nested generating sets for invariant subalgebras. We describe a few applications, in particular, to convergence groups acting on Cantor sets. Under certain finiteness assumptions one can deduce that such actions are necessarily geometrically finite, and hence arise as the boundaries of relatively hyperbolic groups. Similar results have already been obtained by Gerasimov by other methods. One can also use these techniques to give an alternative approach to the Almost Stability Theorem of Dicks and Dunwoody.