The $K(\pi,1)$-conjecture implies the center conjecture for Artin groups

TL;DR

This paper demonstrates that the $K(c6,1)$-conjecture for Artin groups implies the center conjecture, establishing that non-spherical Artin groups satisfying the $K(c6,1)$-conjecture have trivial centers.

Contribution

It proves a new implication between two major conjectures in the theory of Artin groups, linking the $K(c6,1)$-conjecture to the center conjecture.

Findings

Artin groups without spherical factors satisfying the $K(c6,1)$-conjecture have trivial centers.

The $K(c6,1)$-conjecture implies the center conjecture for a broad class of Artin groups.

Establishes a logical dependency between two fundamental conjectures in geometric group theory.

Abstract

In this note, we prove that the $K(\pi,1)$-conjecture for Artin groups implies the center conjecture for Artin groups. Specifically, every Artin group without a spherical factor that satisfies the $K(\pi,1)$-conjecture has a trivial center.