Strict concavity of the growth indicator function for relatively Anosov groups

Let $\Gamma$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank.

The growth indicator function $\psi_\Gamma$ records the directional exponential growth of the Cartan projections of elements of $\Gamma$ in the positive Weyl chamber $\mathfrak a^+$.

We prove that if $\Gamma$ is a non-elementary relatively Borel Anosov group, then $\psi_\Gamma$ is strictly concave on non-collinear directions.

We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $\phi\mapsto\delta^\phi(\Gamma)$.

More generally, for a non-elementary $\theta$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $\theta$-Manhattan hypersurface that is positive on the $\theta$-limit cone and has a critical gap at infinity.

In particular, the $\theta$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $\theta$-Anosov groups, and their $\theta$-growth indicator functions are strictly concave on non-collinear directions.