I am currently reading this paper:http://ac.els-cdn.com/0021869371901074/1-s2.0-0021869371901074-main.pdf?_tid=dbe2268a-086f-11e5-b2dc-00000aacb35d&acdnat=1433171402_aab9d149d19998874901ebf011ab0a5f
And it is not going too well for me.
First of all, what is the map $w$ he defined at the start. I know it should be some sort of way of multiplying things(I think), but this map depends on the word.
Secondly, how can one show for any analytic mapping from a connected analytic manifold $M$ to an analytic manifold $N$, the inverse image of a point in $N$ is either the whole of M or has measure zero in $M$.
Thirdly, what does no relation $w$ is satisfied identically on $G$ mean?
Finally, can someone give me a concrete example of this? (Say $GL(2,Z)$, and $n=2$)
I am only a second year maths undergraduate. I only know basic definitions on Haar measure and Lie group.
Thanks in advance
The map $w$ is defined by the word $w$ as follows. Suppose you have a word in, say, $2$ generators, e.g. $w=aba^2b^3$. This defines an analytic mapping $w\colon G^2\rightarrow G$ by $(a,b)\mapsto aba^2b^3$.
Secondly, this is easily proved by taking $M$ to be an open subset of $\mathbb{R}^n$ and using induction on $m$, and Fubini’s theorem. OK, here you have to fill in some details.
Thirdly, $w\equiv 1$ is satisfied identically in $G$ means $aba^2b^3=1$ for all $a,b$ in our example.
Finally, did you study free groups already ? I think, you should do this first.