Bibliographic Information: Vergara, I. (2024). Property (TTT) and the Cowling–Haagerup constant. arXiv preprint arXiv:2404.00433v2.
Research Objective: This paper investigates the relationship between Property (TTT) in group theory and the concept of weak amenability, particularly when the Cowling–Haagerup constant is 1. The author aims to demonstrate that Property (TTT) obstructs weak amenability under this specific condition.
Methodology: The study employs a theoretical approach, utilizing established concepts and theorems within group theory. It leverages the equivalence of Property (TTT) and Property (TP), as demonstrated by Ozawa and Dumas, to prove the main theorem. The author then applies this result to specific group structures, including semidirect products and lattices in higher rank algebraic groups.
Key Findings: The paper's central finding is that Property (TTT) prevents a group from being weakly amenable when its Cowling–Haagerup constant is 1. This result is significant because it establishes a concrete connection between these important group properties. Furthermore, the paper applies this finding to demonstrate that certain semidirect products and lattices in higher rank algebraic groups cannot be weakly amenable with a Cowling–Haagerup constant of 1.
Main Conclusions: The author concludes that Property (TTT) serves as a strong indicator of the non-amenability of certain groups, particularly when their Cowling–Haagerup constant is 1. This conclusion has implications for understanding the structure and behavior of these groups.
Significance: This research contributes significantly to the field of geometric group theory, specifically in the study of amenability and related properties. It provides a valuable tool for analyzing the structure and properties of groups, particularly those relevant to geometric group theory.
Limitations and Future Research: The paper primarily focuses on theoretical aspects of Property (TTT) and weak amenability. Further research could explore the practical implications of these findings in areas where these group properties are relevant, such as operator algebras and ergodic theory. Additionally, investigating other group properties that might exhibit similar relationships with weak amenability could be a fruitful avenue for future research.
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by Ignacio Verg... at arxiv.org 10-07-2024
https://arxiv.org/pdf/2404.00433.pdfDeeper Inquiries