Rigidity of symmetric simplicial complexes and the lower bound theorem

We show that if Γ is a point group of R^k+1 of order two for some k ≥ 2 and S is a k-pseudomanifold which has a free automorphism of order two, then either S has a Γ-symmetric infinitesimally rigid realisation in R^k+1 or k = 2 and Γ is a half-turn rotation group. This verifies a conjecture made by Klee, Nevo, Novik and Zheng for the case when Γ is a point-inversion group. Our result implies that Stanley's lower bound theorem for centrally symmetric polytopes extends to pseudomanifolds with a free simplicial automorphism of order 2, thus verifying (the inequality part of) another conjecture of Klee, Nevo, Novik and Zheng. Both results actually apply to a much larger class of simplicial complexes - namely, the circuits of the simplicial matroid. The proof of our rigidity result adapts earlier ideas of Fogelsanger to the setting of symmetric simplicial complexes.