Let $$X_n=\{(a,b,c)\in (SU(n))^{3} : c=aba^{-1}b^{-1}\}\subset SU(n)$$ with $SU(n)$ munished with its standard metric (say, normalized so that the total volume is $1$).
Is there a good method to compute the "area" measure of this set? I'm not only looking for the solution to this single problem (though it does interest me), but to a general framework that would allows to consider many more cases (e.g.: other Lie groups, or other "group-like" equations- I don't know if there is a name for such a thing)...
I guess the whole Weyl theory around maximal tori and so on might be useful, but I don't really see how to apply it here.
I must precise also that I think there might be some technic that would consist on going to the Lie algebra, with gaussian measure on it with parameter $t\to \infty$. But this seems heavy and I'm not convincted this simplify the problem...
EDIT: At the view of the replies, I guess I should precise that the measure I want to compute is the one which is not trivial... that is, I don't want to compute the haar measure of my set. At each point of $(SU(n))^3$, I have the canonical $3n$-form $vol$ defined by the volume element. Set $V_x$ the space orthogonal to $X_n$ at $x$, and $n_x$ the unitary form spanning $V_x$ ( unitarity is well-defined by the metric on my Lie group). Then $vol$ can be written as $vol=n\wedge a$, for some $dim(X_n)$-form $a$ defined on $X_n$. What I want to compute is the integrale of $a$ over $X_n$. (here I have acted as if $X_n$ was a smooth submanifold, the singular part being "neglictable").
Hint: If $m$ is Haar measure on a locally compact group $G$ and $S$ is as above then applying Tonelli's theorem to $\chi_S$ shows that $$m\times m\times m(S)=\int_G\int_Gm(\{c\in G:c=aba^{-1}b^{-1}\})\,dadbb=\int_G\int_Gm(\{aba^{-1}b^{1-}\})\,dadb.$$
And if $G$ contains an infinite compact set then the measure of a singleton is ..., because... .