Topological restrictions on relatively Anosov representations

We obtain restrictions on which groups can admit relatively Anosov representations into specified target Lie groups, by examining the topology of possible Bowditch boundaries and how they interact with the Anosov limit maps. For instance, we prove that, up to finite index, any group admitting a relatively Anosov representation into SL(3,R) is a free group or surface group, and any group admitting a relatively k-Anosov representation into Sp(2m,R), where k is an odd integer between 1 and m, is a surface group or a free product of nilpotent groups. We also obtain a characterization of groups admitting relatively 1-Anosov representations into SL(4,R), general bounds on the dimension of the Bowditch boundary of groups admitting relatively Anosov representations into SL(d,R), statements relating spheres in the Bowditch boundary to the (non-)existence of relatively Anosov representations, and a characterization of groups of cohomological dimension at least d-1 admitting relatively 1-Anosov representations into SL(d,R).