Sporadic simple groups and locally toroidal polytopes

Dmitrii V. Pasechnik

Published 2016-03-05Version 1

We construct new finite rank 6 locally toroidal regular polytopes of type {3,3,4,3,3}, disproving a long-standing conjecture of P.McMullen and E.Schulte. These (abstract) polytopes are related to a well-known Y-shaped presentation for the sporadic simple group $Fi_{22}$, and admit (covers of) $O_8^+(2){:}S_3$ as the automorphism group. We also discuss further extensions of these abstract polytopes, and pose a number of open questions.