Groups of type L_2(q) acting on polytopes

TL;DR

This paper classifies certain rank 4 string C-groups isomorphic to L_2(q), showing they only occur for q=11 or 19, corresponding to specific well-known regular 4-polytopes.

Contribution

It proves that L_2(q) can only act as a rank 4 string C-group for q=11 or 19, linking these groups to specific regular 4-polytopes.

Findings

L_2(q) acts as a rank 4 string C-group only for q=11 or 19.

Corresponding polytopes are the 11-cell and 57-cell.

These polytopes are locally projective regular 4-polytopes.

Abstract

We prove that if G is a string C-group of rank 4 and G is isomorphic to L_2(q) with q a prime power, then q must be 11 or 19. The polytopes arising are Grunbaum's 11-cell of type {3,5,3} for L_2(11) and Coxeter's 57-cell of type {5,3,5} for L_2(19), each a locally projective regular 4-polytope.