In a recent paper, Girard uses the geometry of interaction in the hyperfinite
factor in an innovative way to characterize complexity classes. The purpose of
this paper is two-fold: to give a detailed explanation of both the choices and
the motivations of Girard's definitions, and - since Girard's paper skips over
some non-trivial details and only sketches one half of the proof - to provide a
complete proof that co-NL can be characterized by an action of the group of
finite permutations. We introduce as a technical tool the non-deterministic
pointer machine, a concrete model that computes the algorithms represented in
this setting.