[Paper Review] The action dimension of Artin groups

Giang Le|arXiv (Cornell University)|Aug 29, 2016

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper establishes an upper bound of $2n+1$ on the action dimension of Artin groups whose nerve $L$ is an $n$-dimensional simplicial complex, under the $K(/pi,1)$-Conjecture and trivial top cohomology $H^n(L,\mathbb{Z})=0$. The result is proven via a novel construction of an aspherical manifold of dimension $2n+1$ using posets of groups and thickened dual cones, generalizing known results for right-angled and spherical Artin groups.

ABSTRACT

The \emph{action dimension} of a discrete group $G$ is the minimum dimension of a contractible manifold, which admits a proper $G$-action. In this paper, we study the action dimension of general Artin groups. The main result is that the action dimension of an Artin group with the nerve $L$ of dimension $n$ for $n e 2$ is less than or equal to $(2n + 1)$ if the Artin group satisfies the $K(π, 1)$-Conjecture and the top cohomology group of $L$ with $\mathbb{Z}$-coefficients is trivial. For $n = 2$, we need one more condition on $L$ to get the same inequality; that is the fundamental group of $L$ is generated by $r$ elements where $r$ is the rank of $H_1(L, \mathbb{Z})$.

Motivation & Objective

• To determine the action dimension of general Artin groups, particularly those satisfying the $K(\pi,1)$-Conjecture.
• To extend known results on action dimensions of braid groups and right-angled Artin groups to broader classes of Artin groups.
• To establish a sharp upper bound on the action dimension using geometric and topological constructions.
• To resolve the action dimension for Artin groups with spherical subgroups by combining manifold realization and poset group techniques.

Proposed method

• Construct an aspherical manifold of dimension $2n+1$ using a poset of groups over the barycentric subdivision of the nerve $L$.
• Realize the poset of groups via aspherical manifolds associated to each simplex in $L$, with dimension $2i+1$ for $i$-dimensional simplices.
• Use thickened dual cones $\text{Th}D_\sigma$ to glue local manifolds $M_\sigma$ across boundaries, preserving manifold structure.
• Ensure the gluing respects the poset structure and results in a global manifold via disjoint submanifold embeddings and codimension-0 submanifold compatibility.
• Leverage the $K(\pi,1)$-Conjecture to guarantee the universal cover of the constructed manifold is contractible.
• Apply Haefliger’s results on contractibility of basic constructions to confirm the resulting manifold is aspherical and admits a proper group action.

Experimental results

Research questions

• RQ1 What is the action dimension of an Artin group whose nerve $L$ has dimension $n \neq 2$ and satisfies the $K(\pi,1)$-Conjecture with trivial top cohomology?
• RQ2 How does the action dimension change when $n=2$, and what additional condition on $\pi_1(L)$ is required to maintain the same bound?
• RQ3 Can the action dimension of Artin groups be bounded above by $2n+1$ using a geometric construction of an aspherical manifold?
• RQ4 Under what conditions does the action dimension of an Artin group equal $2n+1$, particularly when it contains a spherical subgroup of the same dimension?
• RQ5 Is there a uniform construction method for aspherical manifolds realizing Artin groups that achieves the minimal possible dimension?

Key findings

• The action dimension of an Artin group with nerve $L$ of dimension $n \neq 2$ is at most $2n+1$ if $A_L$ satisfies the $K(\pi,1)$-Conjecture and $H^n(L,\mathbb{Z}) = 0$.
• For $n=2$, the same bound holds if $\pi_1(L)$ is generated by $r$ elements where $r = \text{rk}(H_1(L,\mathbb{Z}))$.
• The construction yields a global aspherical manifold of dimension $2n+1$ that admits a proper action by the Artin group $A_L$, proving the upper bound.
• The action dimension of an irreducible spherical Artin group with $n$-dimensional nerve is exactly $2n+1$, and this value is inherited by any Artin group containing such a subgroup.
• The method generalizes previous results on right-angled Artin groups and braid groups by extending the manifold construction to a broader class of Artin groups.
• The key technical innovation is the use of thickened dual cones in the poset of groups to ensure consistent gluing and manifold structure across local pieces.