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.
