Relatively Anosov representations via flows I: theory

This is the first in a series of two papers that develops a theory of relatively Anosov representations using the original “contracting flow on a bundle” definition of Anosov representations introduced by Labourie and Guichard-Wienhard. In this paper we will mostly focus on general theory while in the second paper we will focus on examples. In the case of relatively hyperbolic groups, this bundle construction involves several choices: the model Gromov-hyperbolic space the group acts on and the norms on the fibers of the bundle. We use the properties of these bundles to define a subclass of nicely behaved relatively Anosov representations, which we call uniformly relatively Anosov. We also prove a stability result.