Overview
This research investigates the embeddability of right-angled Artin groups (RAAGs) into hierarchically hyperbolic groups (HHGs). The primary objective is to define sufficient conditions under which specific powers of a finite collection of elements within an HHG can generate a right-angled Artin subgroup. An additional condition is identified for this generated subgroup to exhibit undistortion.
Research Context
The study operates within the theoretical frameworks of hierarchically hyperbolic groups and right-angled Artin groups. Hierarchically hyperbolic groups encompass a broad class of groups, including mapping class groups, certain groups acting on CAT(0) spaces, and right-angled Artin groups themselves. Right-angled Artin groups are characterized by a presentation with generators and commutativity relations based on a graph structure.
Approach
The research establishes sufficient conditions within a hierarchically hyperbolic group for certain powers of a finite collection of its elements to generate a right-angled Artin subgroup. The conditions for undistortion of this subgroup are also specified. The study then applies these general conditions to two distinct situations:
- One situation is modeled on mapping class groups, where the verification relies on structural assumptions inherent to the ambient hierarchically hyperbolic group itself.
- The second situation is modeled on right-angled Artin groups, requiring the selected elements to be rigidly fully supported.
Findings
The central finding is the identification of sufficient conditions for a finite collection of elements within a hierarchically hyperbolic group, when raised to suitable powers, to generate a right-angled Artin subgroup. Furthermore, an additional condition was found that ensures this generated subgroup is undistorted.
These conditions were specifically verified in two distinct scenarios:
- In scenarios modeled on mapping class groups, the verification was achieved through the application of structural assumptions pertaining to the ambient hierarchically hyperbolic group.
- In scenarios modeled on right-angled Artin groups, the chosen elements were required to be rigidly fully supported for the conditions to hold.
The results of this study recover existing embedding theorems relevant to mapping class groups. Additionally, they extend the non-annular undistortion theorem established by Clay, Leininger, and Mangahas. The work also recovers the Kim–Koberda extension graph theorem specifically for right-angled Artin groups.
Why This Matters
This research contributes to the understanding of the structural relationships between hierarchically hyperbolic groups and right-angled Artin groups. By establishing conditions for the embeddability and undistortion of RAAGs within HHGs, it builds upon and unifies previous findings in geometric group theory. The recovery of known theorems and the extension of others indicate a broader theoretical framework for analyzing these group structures.