In human spatial awareness, information appears to be represented according to 3-D projective geometry. It structures information integration and action planning within an internal representation space. In previous work it has been shown that geometrically constrained active inference can be used as a framework for understanding and modeling central aspects of human spatial consciousness (see the Projective Consciousness Model, PCM). We aim to investigate how the structure of the representation space influences exploration behaviors. The way different perspectives are related and transform a world model defines a specific perception and imagination scheme. In mathematics, such collections of transformations correspond to a ‘group’, whose ‘actions’ characterize the geometry of space. Imbuing world models with a group structure may capture different agents’ spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such a geometric structure impacts agents’ behavior, comparing how Euclidean and projective groups act on epistemic value in active inference and drive curiosity and exploration behaviors. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task, in which the agent is driven solely by curiosity and information gain. The projective group’s nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward an object of interest. The projective group structure within the agent's world model contains the Projective Consciousness Model, which captures key features of consciousness. On the other hand, the Euclidean group had no effect on epistemic value : no action was better than the initial idle state. In structuring a priori an agent’s internal representation space, we show how geometry may in itself induce motion through curiosity, and can play a key role in information integration and action planning.