Core Concepts

Twisted right-angled Artin groups (T-RAAGs) are subgroup separable if and only if their defining mixed graphs do not contain paths or squares on four vertices, a condition equivalent to the subgroup separability of their underlying right-angled Artin groups (RAAGs).

Abstract

Foniqi, I. (2024). Subgroup separability of twisted right-angled Artin groups. *arXiv preprint arXiv:2410.06914*.

This paper aims to characterize the subgroup separability of twisted right-angled Artin groups (T-RAAGs) based solely on their defining mixed graphs.

The author utilizes the properties of subgroup separability, normal forms in T-RAAGs, and the Reidemeister-Schreier procedure to analyze the structure of T-RAAGs and their subgroups. The proof relies on induction on the number of vertices in the defining graph and examines specific cases based on the graph's structure.

- A T-RAAG is subgroup separable if and only if its underlying simplicial graph does not contain induced subgraphs in the form of paths or squares on four vertices (P4 and C4).
- This condition for subgroup separability in T-RAAGs is equivalent to the condition for subgroup separability in their underlying RAAGs.
- The presence of directed edges in the mixed graph, representing Klein bottle relations, does not affect the subgroup separability of the corresponding T-RAAG compared to its underlying RAAG.

The subgroup separability of a T-RAAG can be determined directly from its defining mixed graph by examining the presence of specific induced subgraphs. This result extends previous work on the subgroup separability of RAAGs and provides a complete characterization for the broader class of T-RAAGs.

This research contributes to the understanding of T-RAAGs, a class of groups generalizing RAAGs, and provides a practical method for determining their subgroup separability. This has implications for geometric group theory and related areas where these groups play a significant role.

The paper focuses specifically on subgroup separability and does not explore other algebraic properties of T-RAAGs. Further research could investigate the impact of the characterization on other group-theoretic aspects or explore the connections between T-RAAGs and other classes of groups.

To Another Language

from source content

arxiv.org

Stats

Quotes

Key Insights Distilled From

by Islam Foniqi at **arxiv.org** 10-10-2024

Deeper Inquiries

This characterization provides a powerful tool for studying the geometric and topological properties of T-RAAGs by connecting their algebraic structure (subgroup separability) directly to their defining graphs. Here's how:
Understanding Subgroup Embeddings: Knowing precisely when a T-RAAG is LERF allows us to determine which groups can be embedded in them. This is crucial for understanding the subgroup structure of T-RAAGs and their relationship to other groups. For instance, the absence of induced paths or squares on four vertices in the defining graph guarantees that the corresponding T-RAAG cannot contain "poisonous" subgroups like A(P4) or A(C4), which are known obstructions to LERF.
Geometric Realizations: LERF groups often admit nice geometric realizations. For example, they frequently act properly discontinuously and cocompactly on contractible spaces, leading to classifying spaces with finite skeleta. This connection between LERF and geometric group theory can be exploited to study the geometry of T-RAAGs and their associated spaces.
Residual Properties and Approximations: Subgroup separability implies various residual properties. A group is residually finite if every non-identity element can be separated from the identity in some finite quotient. LERF groups are residually finite, and this residual finiteness has implications for approximating T-RAAGs by finite groups, which can be useful for computational purposes.
Connections to Topology: Subgroup separability plays a significant role in low-dimensional topology, particularly in the study of 3-manifolds. For instance, it's closely related to the virtual Haken conjecture. The characterization of LERF T-RAAGs could potentially shed light on the topological properties of spaces where these groups act, especially in the context of 3-manifold theory.

Yes, it's certainly possible that alternative characterizations exist. Here are some potential avenues for exploration:
Finer Graph Properties: While the presence of induced paths or squares on four vertices is the obstruction to LERF in both RAAGs and T-RAAGs, it might be possible to find more refined graph-theoretic properties that capture subgroup separability in T-RAAGs more precisely. This could involve considering the specific arrangements of directed edges within these graphs.
Group-Theoretic Invariants: Exploring other group-theoretic invariants beyond the existence of specific subgroups could lead to alternative characterizations. For example:
Cohomological Properties: Examining the cohomology groups of T-RAAGs and their relationship to the defining graphs might reveal connections to subgroup separability.
Growth Properties: The growth rate of a group, which measures how the size of balls in its Cayley graph increases, can sometimes be linked to residual properties like LERF. Investigating the growth properties of T-RAAGs could be fruitful.
Complexity of the Word Problem: The complexity of the word problem for a group is another invariant that might be related to its subgroup separability. Exploring this connection in the context of T-RAAGs could be interesting.
Generalizations of Existing Techniques: Techniques used to study subgroup separability in RAAGs, such as those involving graphs of groups or quasi-isometric embeddings, could potentially be adapted and extended to the setting of T-RAAGs.

While this research doesn't directly solve the subgroup membership problem (GMP) for T-RAAGs, it provides valuable insights and potential avenues for further investigation:
LERF Implies Solvable GMP: As noted in Remark 3.3, being LERF implies the solvability of the GMP for finitely presented groups. Therefore, the characterization of LERF T-RAAGs immediately tells us that the GMP is decidable for this subclass.
Focus on Non-LERF Cases: The research highlights the specific graph-theoretic obstructions (induced paths or squares on four vertices) that lead to non-LERF T-RAAGs. This focuses the search for potential algorithms or approaches to tackle the GMP in these more challenging cases.
Potential for Specialized Algorithms: Understanding the structure of T-RAAGs and their connection to graphs could lead to the development of specialized algorithms for solving the GMP, at least for certain subclasses of T-RAAGs.
Insights for Related Groups: The techniques and results from this research could potentially be extended or adapted to study the GMP in groups that are closely related to T-RAAGs, such as graph products of groups or other generalizations of RAAGs.

0