Strict concavity of the growth indicator function for relatively Anosov groups

Let $\mathrm{\Gamma }$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function ${\psi }_{\mathrm{\Gamma }}$ records the directional exponential growth of the Cartan projections of elements of $\mathrm{\Gamma }$ in the positive Weyl chamber ${\mathfrak{a}}^{+}$. We prove that if $\mathrm{\Gamma }$ is a non-elementary relatively Borel Anosov group, then ${\psi }_{\mathrm{\Gamma }}$ 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 $\phi \mapsto {\delta }^{\phi }(\mathrm{\Gamma })$. 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.