Asymptotically large free semigroups in Zariski dense discrete subgroups of Lie groups

Let $G$ be a connected algebraic semisimple real Lie group with finite center and no compact factors, and let $\Gamma$ be a Zariski dense discrete subgroup of $G$. We show that $\Gamma$ contains free, finitely generated subsemigroups whose critical exponents are arbitrarily close to that of $\Gamma$. Furthermore, these subsemigroups are Zariski dense in $G$ and $P$-Anosov in the sense of Kassel--Potrie. This shows that no gap phenomenon holds for critical exponents of discrete subsemigroups of Lie groups, which is in contrast with Leuzinger's critical exponent gap theorem for infinite covolume discrete subgroups of Lie groups with Kazhdan's property (T), proven in 2003. As an application, we prove that the critical exponent is lower semicontinuous in the Chabauty topology, in the following sense: if a sequence of Zariski dense discrete subgroups $\{\Gamma_{n}\}$ of $G$ converges in the Chabauty topology to a Zariski dense discrete subgroup $\Gamma$, then $\liminf_{n \to \infty} \delta(\Gamma_{n}) \geq \delta(\Gamma)$.