Large automorphism groups compared to the p-rank of algebraic curves in characteristic p

Let $\cX$ be a (projective, geometrically irreducible, non-singular) algebraic curve of genus $\ge 2$ and positive $p$-rank $\gamma(\cX)$, defined over an algebraically closed field $\mathbb{K}$ of positive characteristic $p>0$. Contrary to what occurs for the genera, no function $h(\gamma)$ exists such that $|\aut(\cX)|\le h(\gamma)$ whenever $\gamma=\gamma(\cX)$. Thus, to have a bound on $|\aut(\cX)|$ only depending on $\gamma(\cX)$, some restrictions on $\cX$ and $\aut(\cX)$ are needed. In this context, the following theorem is proven. Let $\Gamma$ be a subgroup of $\aut(\cX)$. Assume the existence of a point $P\in \cX$ such that if $S_P$ is the Sylow $p$-subgroup of $\Gamma_P$ fixing $P$, then the quotient curve $\cX/S_P$ is rational. Then %$\gamma(\cX)\ge 2$ and the following $p$-rank analog of the Riemann-Hurwitz bound \begin{equation*} %\label{eq18122025} |\Gamma|<900 \left(\frac{p}{p-1}\right)^4 \gamma(\cX)^4 \end{equation*} holds, unless a subgroup of index $\le 2$ of $\Gamma$ fixes $P$. This bound is sharp apart from the constant.