  A recurring problem in game semantics is to enforce uniformity in strategies.
Informally, a strategy is uniform when the Player's behaviour does not depend
on the particular indexing of moves chosen by the Opponent. In game semantics,
uniformity is used to define a resource modality !, that can be exploited for
the semantics of programming languages.
  In this paper we give a new account of uniformity for strategies on event
structures. This work is inspired by an older idea due to Melli\`es, that
uniformity should be expressed as "bi-invariance" with respect to two
interacting group actions. We explore the algebraic foundations of
bi-invariance, adapt this idea to the language of event structures and define a
general notion of uniform strategy in this context. Finally we revisit an
existing approach to uniformity, and show how this arises as a special case of
our constructions.
