Quasi-K\"ahler groups, 3-manifold groups, and formality

TL;DR

This paper classifies groups that are simultaneously fundamental groups of quasi-Kähler manifolds and 3-manifolds, analyzing their algebraic properties and formality, and explores their geometric and topological implications.

Contribution

It provides a complete classification of such groups at the Malcev level, computes their coranks, and relates corank to the isotropy index, also examining formality in related geometric contexts.

Findings

Classified all 1-formal groups realizable as both quasi-Kähler and 3-manifold groups.

Established that corank equals the isotropy index of the cup-product map.

Described the positive-dimensional components of the characteristic variety for surface singularities.

Abstract

In this note, we address the following question: Which 1-formal groups occur as fundamental groups of both quasi-K\"ahler manifolds and closed, connected, orientable 3-manifolds. We classify all such groups, at the level of Malcev completions, and compute their coranks. Dropping the assumption on realizability by 3-manifolds, we show that the corank equals the isotropy index of the cup-product map in degree one. Finally, we examine the formality properties of smooth affine surfaces and quasi-homogeneous isolated surface singularities. In the latter case, we describe explicitly the positive-dimensional components of the first characteristic variety for the associated singularity link.