Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

We establish several results about Patterson–Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves–Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are $C^1$-regular, from which we deduce that the growth indicator is $C^1$-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim–Oh–Wang. Our methods are dynamical, and we exploit the fact due to Kim–Oh and Blayac–Canary–Zhu–Zimmer that Bowen–Margulis–Sullivan measures for relatively Anosov groups are finite and mixing.