In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible 2-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic 2-dimensional $G$ -complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where $G=\mathrm{PSL}_{2}(2^{n})$ , $G=\mathrm{PSL}_{2}(q)$ with $q\equiv 3\pmod{8}$ or $G=\mathrm{Sz}(2^{n})$ . The cases $G=\mathrm{PSL}_{2}(q)$ with $q\equiv 5\pmod{8}$ are addressed in the second part.
