Flawed groups and the topology of character varieties

TL;DR

This paper introduces the concept of flawed groups, characterizes their properties, and demonstrates that various classes of groups, including free products of nilpotent groups and certain direct products, are flawed, expanding the known examples.

Contribution

It defines flawed groups and proves that free products of nilpotent groups, certain RAAGs, and some direct products are flawed, unifying and extending previous classifications.

Findings

All finitely generated free products of nilpotent groups are flawed.

RAAGs with torsion are conjectured to be flawed.

Direct products of finite groups with some flawed groups are also flawed.

Abstract

A finitely presented group F is called flawed if Hom(F,G)//G deformation retracts onto its subspace Hom(F,K)/K for reductive affine algebraic groups G and maximal compact subgroups K in G. After discussing generalities concerning flawed groups, we show that all finitely generated groups isomorphic to a free product of nilpotent groups are flawed. This unifies and generalizes all previously known classes of flawed groups. We also provide further evidence for the authors' conjecture that RAAGs (with torsion) are flawed. Lastly, we show direct products between finite groups and some flawed group are also flawed. These latter two theorems enlarge the known class of flawed groups.