Strict concavity of the growth indicator function for relatively Anosov groups

Abstract

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 $\varphi \mapsto {\delta }^{\varphi }(\Gamma )$. 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.

31 pages. Comments welcome!