Quantum Computation By Adiabatic Evolution

I'm having trouble understanding some parts of that article (Quantum Computation By Adiabatic Evolution, by Farhi, Goldstone, Gutmann and Sipser, 2000).

The article describes a different model for quantum computation (in 2004 it was proved to be equivalent to the standard model), and an analysis for some simple problems.

I had problem understanding the following points:
1. On page 9, they write "The alert reader may have noticed that two of the levels in fig. 6 cross. this can be understood in terms of a symmetry."
Is this phenomena general? can there be a crossing of the eigenvalues without a symmetry? I can't understand why not.
2. On page 14, they write: "Because H(s) is invariant under the translation b_j -> b_{j+1} ...
what does that mean?
let A: b_j -> b_{j+1} . this operator isn't linear, since the state |+> ( 1/2^0.5 (|0>+|1>) ) is in the kernel of sigma_minus, and therefore in
the kernel of b_j , and not necessarily of b_{j+1}.
did they mean that H(s)=A^-1*H(s)*A ?
3. On page 14, the say that because that H(s) is invariant under the translation described in 2. and that H(s) is quadratic in b_j and b_j*, a transformation to fermion operators would enable a description of H(s) as a sum of commuting operators.
Why?
It seems that all these transitions are treated as "common knowledge". I would be happy to get some relevant references for that (I am capable of understanding some physics text).
4. Further in page 14, they say "let |Omega_p> be the state annihilated by both beta_p and beta_{-p}, that is beta_p|Omega_p>=beta_{-p}|Omega_p>=0. "
Why does such a state exist? I didn't understand that paragraph at all.

Any help would be appreciated.

Or Sattath
you can reach me by email at sattath go cs go huji go ac go il
replace go by dot.
.