Root distributions in Moebius-Kantor complexes

TL;DR

This paper investigates the distribution of rank 2 roots in nonpositively curved 2-complexes with Moebius--Kantor links, revealing constraints on parity distributions and classifying root distributions in certain cases.

Contribution

It introduces the concept of parity distributions in Moebius--Kantor complexes, shows their restrictions, and classifies root distributions for even parity cases, including the unique Pauli complex.

Findings

Certain parity distributions are impossible in Moebius--Kantor complexes.

All root distributions can be realized despite parity restrictions.

The Pauli complex is the unique even simply connected Moebius--Kantor complex.

Abstract

We study the distribution of roots of rank 2 in nonpositively curved 2-complexes with Moebius--Kantor links. For every face in such a complex, the parity of the number of roots of rank 2 in a neighbourhood of the face is a well-defined geometric invariant determined by the root distribution. We study the relation between the root distribution and the parity distribution. We prove that there exist parity distributions in flats which are disallowed in Moebius--Kantor complexes. This contrasts with the fact that every root distribution can be realized. We classify the root distributions associated with an even parity distribution (i.e., such that every face is even) on a flat plane. We prove that there exists up to isomorphism a unique even simply connected Moebius--Kantor complex -- namely, the Pauli complex.