Can a limit of degenerate two-cocycles be non-degenerate?

T

Tron

Let $G$ be a discrete abelian group and $\omega\colon G\times G\to\mathbb{T}$ a two-cocycle on $G$. We say that $\omega$ is non-degenerate if for every $e\neq g\in G$ there exists $h\in G$ such that $\omega(g,h)\neq 1$.

Let $G=\oplus_{i=1}^\infty\mathbb{Z}$, $\omega$ a two-cocycle on $G$, and denote by $\omega_n$ the restriction of $\omega$ to $\mathbb{Z}^n=\oplus_{i=1}^n\mathbb{Z}$.

My question is fairly simple: if $\omega$ is non-degenerate and $m\in\mathbb{N}$, does there always exist some $n\geq m$ such that $\omega_n$ is non-degenerate?
 

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