I was trying to solve an exercise from a book but i got stuck in my calculation.
I'm trying to show that foundamental groups of a class of orientable 3-manifold is not left orderable where by this i mean that they admits a strict ordere invariant for left multiplication.
The groups are Seifert manifold built over a surface where the surface is $\mathbb{R}P^{2}.$
They admit the following presentation: $$\pi_{1}(M)=\langle\gamma_{1},...,\gamma_{n},y,h\;|$$ $$yhy^{-1}=h^{-1},\gamma_{j}^{\alpha_{j}}=h^{-\beta_{j}},\gamma_{j}h\gamma_{j}^{-1}=h,y^{2}\gamma_{1}\cdots\gamma_{n}=1\rangle.$$
With some calculation I got to show that if $k>0,h>1$ the following holds: $h^{-k}<y^2<h^k$ and $h^{-k}<y^{-2}<h^{k}.$ In the book I was given the hint to show that $$yhy^{-1}>1\tag{$*$}$$ and this in facts concludes as it is a contradiction with the assumption that $h>1$ given the first relation but I could not find a way to prove the key fact $(*)$. Any suggestion in what kind of relations I should look for?
Thanks in advance.