  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.
