Quasi-isometric classification of some high dimensional right-angled Artin groups (0906.4519v2)

Abstract: In this note we give the quasi-isometry classification for a class of right angled Artin groups. In particular, we obtain the first such classification for a class of Artin groups with dimension larger than 2; our families exist in every dimension.

• The paper introduces a novel quasi-isometric classification method for high-dimensional RAAGs using bisimilarity on associated trees.
• It extends RAAG classification beyond dimension two by showing that trees from simplicial complexes determine quasi-isometric equivalence.
• The results have practical implications for understanding free product decompositions and the coarse geometry of complex algebraic groups.

Quasi-Isometric Classification of Certain High-Dimensional Right-Angled Artin Groups

The paper of quasi-isometric (QI) classifications has been pivotal in understanding the broader geometric properties of groups and manifolds. This paper focuses on providing a detailed quasi-isometric classification of certain families of right-angled Artin groups (RAAGs) that extend beyond the previously studied dimensions, specifically addressing groups of dimension greater than two.

Background and Motivation

Right-angled Artin groups are versatile mathematical structures defined via finite graphs, where the vertices represent generators, and commutations between these generators are explicitly prescribed by the presence of edges. They serve as interpolations between completely free groups and fully abelian groups. Previous work established the QI classification for RAAGs whose defining graphs are either trees or atomic graphs. The significance lies in RAAGs' ability to capture an intricate balance between algebraic and geometric properties, especially when investigating large-scale geometric properties such as quasi-isometry.

Main Contributions

The authors introduce a novel classification method for RAAGs represented by higher-dimensional simplicial complexes, notably those extending beyond cohomological dimensions of two. The core of the paper involves associating each RAAG with a tree (denoted as $\Gamma(K)$) that is colored based on the graph structure of the complexes. They defined a bisimilarity relation on these trees, providing a novel approach for determining when two RAAGs are quasi-isometrically equivalent.

Key Results

1. Theorem on Quasi-Isometric Classification: A main result demonstrates that for any $K, K' \in \mathcal{T}_n$ (where $\mathcal{T}_n$ represents a specific class of simplicial complexes), the RAAGs $A_K$ and $A_{K'}$ are quasi-isometric if and only if their associated trees $\Gamma(K)$ and $\Gamma(K')$ are bisimilar, up to a permutation within the group of $n+1$ elements. This theorem extends the previous knowledge about QI classifications to encompass a richer, higher-dimensional landscape of RAAGs.
2. Corollary on Maximally Branched RAAGs: The work extends the understanding of maximally branched RAAGs to higher dimensions, stating that any maximally branched RAAGs that are also irreducible are quasi-isometric, regardless of their initial configurations.
3. Implications for Free Products: Another compelling corollary discusses the quasi-isometric invariance in terms of free product decompositions. This result has valuable implications for understanding the broader algebraic structures within groups that essentially preserve their geometric properties under a free product operation.

Implications and Future Directions

The implications of this research are twofold: It enriches the theoretical framework for studying RAAGs by expanding the known QI classifications while also proposing tools such as bisimilarity relations, which could be applied in broader mathematical contexts. This work may inspire further exploration into the commensurability problems associated with such groups, unearthing deeper connections between algebraic properties and their geometric manifestations.

Potential future avenues include applying these methods to other classes of Artin groups with analogous constructions or leveraging the insights into practical applications where understanding the coarse geometry of groups is beneficial.

In conclusion, this paper marks a substantial progression in the field of geometric group theory, particularly in the understanding of high-dimensional RAAGs. Through new classifications and systematic approaches, it sets the stage for ongoing research and exploration within and beyond the field of Artin groups.