Computing the Thurston Norm for Coherent Right-Angled Artin Groups Using L2-Invariants

Core Concepts

This paper introduces a group-theoretic analog of the Thurston semi-norm for coherent right-angled Artin groups, demonstrating that it can be calculated using L2-invariants, specifically the L2-Euler characteristic.

Abstract

• Bibliographic Information: Kudlinska, M. (2024). Thurston norm for coherent right-angled Artin groups via L2-invariants [Preprint]. arXiv:2411.02516v1.
• Research Objective: This paper aims to define a group-theoretic analog of the Thurston semi-norm for coherent right-angled Artin groups and demonstrate a method for its computation using L2-invariants.
• Methodology: The paper utilizes concepts from geometric group theory, particularly Bass-Serre theory, L2-invariants like L2-Betti numbers and L2-Euler characteristics, and the theory of polytopes. It leverages the structure of coherent right-angled Artin groups and their connection to chordal flag complexes.
• Key Findings: The study establishes that for a one-ended coherent right-angled Artin group G, the splitting complexity along an epimorphism to Z equals the L2-Euler characteristic of the kernel. This result allows for the definition of a Thurston-type semi-norm on the first cohomology group of G, mirroring the Thurston norm in 3-manifold theory.
• Main Conclusions: The paper concludes that coherent right-angled Artin groups admit a Thurston-type norm analogous to the Thurston norm in 3-manifold theory. This norm measures the splitting complexity of integral characters and can be computed using L2-invariants.
• Significance: This research significantly contributes to the understanding of coherent right-angled Artin groups by introducing a new invariant and providing a computational method using L2-invariants. It bridges concepts from 3-manifold theory to group theory, potentially opening new avenues for research in both fields.
• Limitations and Future Research: The paper primarily focuses on coherent right-angled Artin groups. Further research could explore extending these results to other classes of groups, such as strongly coherent groups or free-by-cyclic groups. Investigating the properties and applications of this newly defined Thurston-type semi-norm in group theory would also be a promising direction.

Stats

The L2-polytope of the infinite cyclic group Z is represented by the unit interval between 0 and 1 in R.
The L2-polytope of the free abelian group of rank k is trivial for all k ≥ 2.

Quotes

"In this paper we propose a group-theoretic analogue of the Thurston semi-norm which measures the splitting complexity of integral characters of a group."
"The purpose of this article is to show that coherent right-angled Artin groups admit a Thurston-type norm which is completely analogous to the Thurston norm in the setting of 3-manifolds."

Key Insights Distilled From

Thurston norm for coherent right-angled Artin groups via $L^2$-invariants

by Monika Kudli... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02516.pdf

Deeper Inquiries

How does the understanding of the Thurston norm for coherent right-angled Artin groups inform the study of their geometric and topological properties?

Answer:
The Thurston norm, originally defined for 3-manifolds, provides a powerful tool to study the geometric and topological properties of spaces by understanding the complexity of embedded surfaces representing homology classes. The introduction of an analogous Thurston-type semi-norm for coherent right-angled Artin groups (RAAGs), as presented in the paper, similarly unlocks a deeper understanding of their structure and properties. Here's how:

Splitting Complexity and Geometric Decompositions: The Thurston semi-norm directly measures the splitting complexity of the group along integral characters. This provides a quantitative way to analyze how a coherent RAAG can be decomposed into simpler pieces along subgroups. This is analogous to understanding how a 3-manifold can be cut along surfaces.
Algebraic Fiberings and Fibrations: The connection between the Thurston semi-norm and the Euler characteristic of kernels of algebraically fibered epimorphisms (Theorem 1.1) provides a bridge between the algebraic structure of the group and potential fibrations of spaces on which the group acts. This is particularly relevant to understanding the geometric group theory of coherent RAAGs.
Comparison with 3-Manifold Theory: The strong analogy between the Thurston semi-norm for coherent RAAGs and the classical Thurston norm for 3-manifolds suggests a deeper connection between these objects. This opens avenues for applying techniques and insights from 3-manifold theory to the study of coherent RAAGs, and vice versa.
New Invariants and Classifications: The Thurston semi-norm, being a geometrically meaningful invariant, can potentially be used to distinguish between different coherent RAAGs or to classify them based on the properties of their semi-norms.
In summary, the Thurston semi-norm provides a new lens through which to study the interplay between the algebraic structure of coherent RAAGs, their geometric decompositions, and potential topological realizations. This opens up exciting new research directions in geometric group theory and low-dimensional topology.

Could there be alternative methods for computing the Thurston-type semi-norm for coherent right-angled Artin groups that do not rely on L2-invariants?

Answer:
While the paper leverages the powerful machinery of L2-invariants and the L2-polytope to define and compute the Thurston-type semi-norm for coherent RAAGs, it's natural to wonder if alternative approaches exist. Here are some possibilities and challenges:
1. Direct Combinatorial Methods:

Challenge: The definition of splitting complexity involves considering all possible graph-of-groups splittings dual to a given character, which can be a very large and intricate search space.
Potential: Coherent RAAGs have a strong combinatorial structure coming from their defining graphs. It might be possible to exploit this structure to develop combinatorial algorithms for constructing minimal complexity splittings, leading to a direct computation of the semi-norm.
2. Geometric Realizations and Complexity:

Potential: Every RAAG acts freely and cocompactly on a CAT(0) cube complex. One could try to relate the splitting complexity of the group to geometric properties (e.g., minimal surface area) of corresponding objects in the cube complex.
Challenge: The relationship between algebraic splittings of a RAAG and geometric splittings of its associated cube complex is not fully understood.
3. Heegaard-like Splittings:

Inspiration: In 3-manifold theory, Heegaard splittings provide a powerful way to decompose a 3-manifold into simpler pieces.
Potential: One could explore analogous notions of "Heegaard-like" splittings for cube complexes, which might lead to a more geometric and potentially computable way to understand splitting complexity.
Challenge: Defining and working with such splittings in the context of non-positively curved cube complexes would require significant new ideas.
4. Connections to Other Invariants:

Potential: Explore relationships between the Thurston semi-norm and other known invariants of RAAGs, such as their homology groups, cohomology rings, or finiteness properties of subgroups. Such connections might lead to alternative computational methods.
In conclusion, while L2-invariants provide a powerful framework for the Thurston semi-norm, exploring alternative approaches based on the combinatorial and geometric properties of coherent RAAGs is a promising avenue for future research.

What are the implications of the close relationship between the algebraic structure of coherent right-angled Artin groups and the combinatorial structure of their defining graphs in a broader mathematical context?

Answer:
The close interplay between the algebraic structure of coherent right-angled Artin groups (RAAGs) and the combinatorial structure of their defining graphs has profound implications across various mathematical disciplines:
1. Geometric Group Theory:

Model Groups: Coherent RAAGs serve as a rich class of model groups for studying broader phenomena in geometric group theory. Their combinatorial nature makes them amenable to explicit computations and constructions, providing insights into properties like coherence, subgroup distortion, and quasi-isometric rigidity.
CAT(0) Cube Complexes: The action of RAAGs on CAT(0) cube complexes provides a crucial link between their algebraic and geometric properties. This connection has led to significant advances in understanding the structure of these groups and their subgroups.
2. Low-Dimensional Topology:

Special Cube Complexes: Coherent RAAGs provide a bridge between group theory and the study of special cube complexes, which are non-positively curved spaces with rich combinatorial structure. This connection has led to new constructions and classifications of such complexes.
Generalizations of 3-Manifold Techniques: The analogy between the Thurston semi-norm for coherent RAAGs and the classical Thurston norm for 3-manifolds suggests the possibility of extending techniques and ideas from 3-manifold topology to higher dimensions.
3. Combinatorics and Graph Theory:

Flag Complexes and Graph Properties: The study of coherent RAAGs motivates research on flag complexes and their combinatorial properties. For example, understanding which graph properties ensure coherence or other algebraic properties of the corresponding RAAG is an active area of research.
Graph Algorithms and Complexity: The combinatorial nature of RAAGs makes them suitable for algorithmic analysis. Developing efficient algorithms for problems like the word problem, conjugacy problem, or isomorphism problem for these groups has implications for both group theory and theoretical computer science.
4. Connections to Other Fields:

Representation Theory: The structure of RAAGs has connections to the representation theory of Lie algebras and quantum groups.
K-Theory and Algebraic Topology: L2-invariants and the L2-polytope connect the study of RAAGs to K-theory and algebraic topology, providing new tools and perspectives.
In conclusion, the deep relationship between the algebra of coherent RAAGs and the combinatorics of their defining graphs has far-reaching implications. It fosters a fruitful exchange of ideas and techniques between diverse mathematical fields, leading to new discoveries and a deeper understanding of fundamental structures.