The notion of the "equator" in $SU(2)$ can indeed be interpreted in terms of conjugacy classes, which are important in the context of Lie groups and their representations.

Conjugacy Classes in $SU(2)$

A conjugacy class in a group $G$ is the set of elements that are conjugate to each other, meaning for $g\in G$:

$$\text{Conjugacy class of }g=\{hg{h}^{-1}:h\in G\}.$$

Conjugacy Classes in $SU(2)$

For $SU(2)$, the elements are $2\times 2$ unitary matrices with determinant 1. Any element $U\in SU(2)$ can be written as:

$$U=\left(\begin{array}{cc}\alpha & -{\beta }^{\ast }\\ \beta & {\alpha }^{\ast }\end{array}\right),\text{where}|\alpha {|}^2+|\beta {|}^2=1.$$

Conjugacy classes in $SU(2)$ are determined by the trace of the matrix, which is invariant under conjugation. The trace $\text{Tr}(U)$ is given by:

$$\text{Tr}(U)=\alpha +{\alpha }^{\ast }=2\mathrm{\Re }(\alpha ).$$

Since $\alpha =a+bi$ (with $a^2+b^2+|\beta {|}^2=1$), the trace becomes:

$$\text{Tr}(U)=2a.$$

Equator as a Conjugacy Class

The "equator" of $SU(2)$, as we interpreted geometrically, can be linked to a specific value of the trace. For example, if we consider matrices with $a=0$, then:

$$\text{Tr}(U)=0.$$

These matrices form a conjugacy class where the trace is zero. This subset of $SU(2)$ corresponds to matrices that can be written in the form:

$$U=\left(\begin{array}{cc}bi& -{\beta }^{\ast }\\ \beta & -bi\end{array}\right),\text{where}{b}^2+|\beta {|}^2=1.$$

This forms a lower-dimensional manifold within $SU(2)$ and can be visualized as the "equator" on the 3-sphere $S^3$.

Conclusion

The "equator" in $SU(2)$ corresponds to a conjugacy class with a specific trace value. In this case, matrices with a trace of zero form a conjugacy class that geometrically represents a 2-sphere $S^2$ within the 3-sphere $S^3$. This interpretation ties the geometric notion of the equator to the algebraic structure of conjugacy classes in $SU(2)$.