Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.00445

The $K(\pi, 1)$ conjecture for affine Artin groups

Giovanni Paolini·Mario Salvetti
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable under orthogonal reflections are aspherical. We survey three facets of the argument: the combinatorics of noncrossing partition posets associated with Coxeter groups; the appearance of dual Artin groups and the question of their isomorphism with standard Artin groups; the topological models and their interplay in the proof.

Explore related subjects

math.GRmath.ATmath.COmath.GT

Keep this discovery

BibTeXRIS

Giovanni Paolini, Mario Salvetti. 2025-08-30. The $K(\pi, 1)$ conjecture for affine Artin groups. https://arxiv.org/abs/2509.00445

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗