Compactness Gives a Finite SU(2) Base Net

The ambient space $M_2(\mathbb{C})$ is finite-dimensional over $\mathbb{C}$. For $U\in SU(2)$, norm preservation gives $\mathrm{\parallel }U\mathrm{\parallel }=1$ by [F2], so $SU(2)$ lies in the closed operator-norm unit ball. It is closed there because $U^{\ast }U=I$ and $\det U=1$ are continuous equations in the matrix entries and their conjugates. [F1] makes the closed unit ball compact; therefore its closed subset $SU(2)$ is compact.

Let $S$ be the values of finite words in $G$ and let $H=\bar{S}\subseteq SU(2)$. Since $S$ is closed under multiplication and matrix multiplication is continuous, $H$ is a closed semigroup. For each integer $n\ge 1$, let $N_n$ be the least number of operator-norm balls of radius $1\mathrm{/}n$ covering $SU(2)$; compactness gives such a finite cover. Fix $g\in G$. Two of $I,g,\dots ,g^{N_n}$ lie in one cover ball, say $g^r,g^s$ with $r<s$. By [F2], $\mathrm{\parallel }{g}^{s-r}-I\mathrm{\parallel }=\mathrm{\parallel }(g^s-g^r)g^{-r}\mathrm{\parallel }<2\mathrm{/}n$. Let $d_n$ be the least $d\in \{1,\dots ,N_n\}$ with $\mathrm{\parallel }{g}^d-I\mathrm{\parallel }<2\mathrm{/}n$. If $d_n=1$ for infinitely many $n$, then $g=I$ and $g^{-1}=g\in S$; otherwise $d_n\ge 2$ eventually, and $g^{d_n-1}\in S$ satisfies $\mathrm{\parallel }{g}^{d_n-1}-g^{-1}\mathrm{\parallel }=\mathrm{\parallel }(g^{d_n}-I)g^{-1}\mathrm{\parallel }<2\mathrm{/}n$, so $g^{-1}\in H$. This holds for every $g\in G$. Hence $H$ contains $G$ and its inverses, so it contains $⟨G⟩$; closedness of $H$ and density of $⟨G⟩$ give $H=SU(2)$. Therefore word values in $G$ are dense.

For each $U\in SU(2)$, density supplies a word value $g_U\in S$ with $\mathrm{\parallel }U-g_U\mathrm{\parallel }<\delta \mathrm{/}2$. The open balls of radius $\delta \mathrm{/}2$ centered at these word values cover $SU(2)$. Compactness gives a finite subcover with centers $g_1,\dots ,g_r$. Choose one finite word representing each $g_j$; the resulting list is a $\delta$-net, and the maximum of its finitely many word lengths is finite. The compactness argument supplies no effective search cutoff or certificate.