I am trying to study the commutator map of a given Lie group $G$: $$\mu : G\times G\to G,\ \mu(x,y)=[x,y]=xyx^{-1}y^{-1}$$ I am interested in:
- Its singular points (where the the differential is not onto).
- Its fibers, are connected or not?
After some long computations, I get: $$D_{(x,y)}\mu(u,v)=D_xR_{yx^{-1}y^{-1}}(u)-D_x\left(L_{xyx^{-1}}\circ R_{x^{-1}y^{-1}}\right)(u)+D_y\left(L_x\circ R_{x^{-1}y^{-1}}\right)(v)-D_y\left(L_{xyx^{-1}y^{-1}}\circ R_{y^{-1}}\right)(v)$$ Where $L_x$ and $R_x$ are the left and right translations of $G$.
Apart from the obvious singular point $(e,e)$, I don't see how to find the singular points in the cases: $G=SL(2), SU(2), GL(2)$?
Is there a nice reference where this map is studied in detail?
Same question for the map: $$c(x_1,...x_{2g})=[x_1,x_2]...[x_{2g-1},x_{2g}]$$