One-relator Kaehler groups

TL;DR

This paper characterizes exactly which one-relator groups are Kähler, showing they are either finite cyclic or fundamental groups of certain orbifold Riemann surfaces, thus linking algebraic and geometric properties.

Contribution

It provides a complete classification of one-relator Kähler groups, identifying their structure as either cyclic or specific orbifold surface groups.

Findings

One-relator Kähler groups are either finite cyclic or orbifold surface groups.

The classification is complete and explicit.

The result connects algebraic group properties with geometric structures.

Abstract

We prove that a one-relator group $G$ is K\"ahler if and only if either $G$ is finite cyclic or $G$ is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus $g > 0$ with at most one cone point of order $n$: $$< a_1\, b_1\, \,...\, a_g\, b_g\, \mid\, (\prod_{i=1}^g [a_i\, b_i])^n>\, .$$