Given an expander graph family (an injective sequence of graphs with uniformly bounded vertex degree and a Cheeger constant/Laplacian spectral gap uniformly bounded away from zero).
Can the corresponding clique complexes (flag complexes, i.e., all cliques are taken as simplices) all be simply connected? If not, can the clique complexes have uniformly upper bounded first Betti number?
My intuition is that expanders are roughly finite trees, where you magically wrap up the leaves, and this wrapping up might cause a large first Betti number. Another maybe bogus intuition is that expanders are negatively curved and should have large Betti number by a Gauss-Bonnet like argument.
I did not find anything related in the literature after a brief search, so I expect that examples of expanders with simply connected clique complexes should be known. However, I am not very familiar with the standard expander constructions, so I am hoping that the community can help.
Thank you very much!