Locally toroidal polytopes of rank 6 and sporadic groups

TL;DR

This paper expands the classification of finite universal locally toroidal regular polytopes of a specific type, disproves a long-standing conjecture, and links these structures to sporadic groups and orthogonal groups over GF(2).

Contribution

It provides a revised list of such polytopes, introduces a new universal polytope related to Fi22, and explores their connections to Y-shaped presentations and orthogonal groups.

Findings

Disproves a long-standing conjecture about these polytopes.

Identifies a new universal polytope related to the sporadic group Fi22.

Connects known polytopes to Y-shaped presentations of orthogonal groups.

Abstract

We augment the list of finite universal locally toroidal regular polytopes of type {3,3,4,3,3} due to P.McMullen and E.Schulte, adding as well as removing entries. This disproves a related long-standing conjecture. Our new universal polytope is related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admits $S_4\times O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of its quotients in the context of Y-shaped presentations. As well, we note that two known examples of finite universal polytopes of type {3,3,4,3,3} are related to Y-shaped presentations of orthogonal groups over GF(2). Mixing construction is used in a number of places to describe covers and 2-covers.