Growth indicators and conformal measures for transverse subgroups.

Let $G$ be a connected semisimple real algebraic group. Let $\theta$ be a non-empty subset consisting of simple roots of $G$. The class of $\theta$-transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups, $\theta$-Anosov subgroups and their relative versions. For any Zariski dense $\theta$-transverse subgroup $\Gamma$, we introduce  the notion of $\theta$-growth indicators and  discuss their properties and roles in the study of conformal measures,  extending the work of Quint (2003). We also prove that for any $(\Gamma,\psi)$-conformal measure on the $\theta$-boundary,   the conical set of $\Gamma$ has measure either $1$ or $0$, depending on whether the $\psi$-Poincare series  diverges or not; this extends recent works of Sambarino and of Canary-Zhang-Zimmer  proved for special measures supported on the limit set. Our work is new even for $\theta$-Anosov subgroups and answers a question of Sambarino (2022).  Applications include an analogue of the Ahlfors measure conjecture: the limit set of a $\theta$-Anosov subgroup is either the whole boundary or of Lebesgue measure zero. When theta is the set of all simple roots, these were previously obtained by Minju Lee-Oh.