Overview
This research presents an algebraic criterion applicable to specific sequences of groups, $(G_n)_{n\in\mathbb{N}}$, to determine their eventual possession of Kazhdan's property (T). Utilizing this criterion, the study demonstrates that the Brin--Thompson groups $nV$ acquire Kazhdan's property (T) when $n$ is large. The developed algebraic criterion offers a uniform method for establishing property (T) for large $n$ across several group families: $nV$, $\mathrm{Aut}(F_n)$, and $\mathrm{EL}_n(R)$ for a finitely generated associative unital ring $R$. An appended section, authored by Francesco Fournier-Facio, further details properties of $nV$ for large $n$, indicating it is neither hyperlinear nor $\mathrm{MF}$, and is not Schatten $p$-approximable for any $1\le p \infty$.
Research Context
The investigation focuses on Kazhdan's property (T), a significant algebraic characteristic in group theory. The study addresses this property for various group sequences, including the Brin--Thompson groups $nV$, the automorphism groups of free groups $\mathrm{Aut}(F_n)$, and elementary linear groups $\mathrm{EL}_n(R)$ over a finitely generated associative unital ring $R$. The establishment of a unified algebraic criterion for these diverse groups suggests a common underlying structure that influences the presence of property (T) as $n$ increases.
Approach
The methodology involved the development of an algebraic criterion. This criterion was designed to determine if sequences of groups $(G_n)_{n\in\mathbb{N}}$ eventually satisfy Kazhdan's property (T). This approach was then applied to specific group families. The research also included an auxiliary analysis, documented in an appendix, which explored additional characteristics of the Brin--Thompson groups $nV$ for large $n$. This appendix investigated properties such as hyperlinearity, being $\mathrm{MF}$, and Schatten $p$-approximability.
Findings
- An algebraic criterion was established that predicts whether certain sequences of groups, specifically $(G_n)_{n\in\mathbb{N}}$, eventually possess Kazhdan's property (T).
- The Brin--Thompson groups $nV$ were shown to have Kazhdan's property (T) for large $n$.
- The algebraic criterion provides a uniform proof demonstrating that $nV$, $\mathrm{Aut}(F_n)$, and $\mathrm{EL}_n(R)$ (where $R$ is a finitely generated associative unital ring) all exhibit property (T) for large $n$.
- For large $n$, the group $nV$ is neither hyperlinear nor sofic.
- For large $n$, the group $nV$ is not $\mathrm{MF}$.
- For large $n$, the group $nV$ is not Schatten $p$-approximable for any $1\le p \infty$.
Why This Matters
The establishment of a uniform algebraic criterion for Kazhdan's property (T) simplifies the analysis of this property across several distinct families of groups. The specific findings regarding the Brin--Thompson groups, $\mathrm{Aut}(F_n)$, and $\mathrm{EL}_n(R)$ contribute to the understanding of their algebraic structures. The additional characterizations of $nV$ (non-hyperlinearity, non-sofic, non-$\mathrm{MF}$, and not Schatten $p$-approximable) further delineate the algebraic and analytical properties of these groups in the large $n$ regime.