Odd vanishing cycles in cyclotomic fields

TL;DR

This paper explores the structure of odd vanishing cycles, lifts of cusps in Hecke groups, within cyclotomic fields, revealing their orbit structure under a monodromy group and connecting to Coxeter groups.

Contribution

It introduces the concept of odd vanishing cycles and analyzes their orbit structure under a monodromy group, linking to Coxeter groups and providing both research and survey insights.

Findings

Lifts of cusps form discrete subsets in complex plane

Orbit structures under monodromy group are classified

Connections to Coxeter and dihedral groups are established

Abstract

A cusp of a Hecke group $G_q$ has two natural lifts to the ring of integers of a cyclotomic field. These lifts are called here odd vanishing cycles. All lifts of all cusps together form a discrete subset of ${\mathbb C}$ of some exquisite beauty. They form one or two or four orbits of a certain subgroup of the matrix Hecke group. The subgroup can be considered as a monodromy group and is an analog of a rank 2 Coxeter group, so of a dihedral group. The paper has a research part and a larger survey part.