An on-going line of investigation is to determine which simple (or near simple) groups are automorphism groups of abstract regular polytopes, in the sense of the monograph with that title by the reviewer and E. Schulte [Abstract regular polytopes. Cambridge: Cambridge University Press (2002; Zbl 1039.52011)]. Recall that one can identify an abstract regular polytope of rank \(n\) with its automorphism group \(G\), which is a string C-group of rank \(n\), whose generators \(\rho_0,\ldots,\rho_{n-1}\) are involutions satisfying the intersection property \[ \langle \rho_i \mid i \in J \rangle \cap \langle \rho_i \mid i \in K \rangle = \langle \rho_i \mid i \in J \cap K \rangle \] for \(J,K \subseteq \{0,\ldots,n-1\}\); moreover, \(\rho_j,\rho_k\) commute if \(|j-k| \geq 2\).
In this paper, the authors consider the projective special linear group \(G = \mathrm{PSL}(4,\mathbb{F}_q)\) (with \(q = p^k\) a prime power). They prove two main results. First, \(G\) is a string group of rank \(4\) for every \(k\) if the prime \(p\) is odd. More specifically, the Schläfli type of \(G\) is \([q-1,\tfrac12(q^2 + 1),e]\), where \(e = p\) or \(e\) divides \(q \pm 1\), which means that \(\rho_{j-1}\rho_j\) has period \(q-1\), \(\tfrac12(q^2 + 1)\) or \(e\) as \(j = 1,2,3\). Second, \(G\) is not a string C-group of any rank for any \(k\) if \(p = 2\).