The quotient of a Hermitian symmetric space of non-compact type by a torsion-free cocompact arithmetic subgroup of the identity component of the group of isometries of the symmetric space is called an arithmetic fake compact Hermitian symmetric space if it has the same Betti numbers as the compact dual of the Hermitian symmetric space. This is a natural generalization of the notion of “fake projective planes” to higher dimensions. Study of arithmetic fake compact Hermitian symmetric spaces of type An with even n has been completed in [PY1], [PY2]. The results of this paper, combined with those of [PY2], imply that there does not exist any arithmetic fake compact Hermitian symmetric space of type other than An, n ≤ 4 (see Theorems 1 and 2 in the Introduction below and Theorem 2 of [PY2]). The proof involves the volume formula given in [P], the Bruhat-Tits theory of reductive p-adic groups, and delicate estimates of various number theoretic invariants.
Gopal PRASAD. Sai-Kee YEUNG. "Nonexistence of arithmetic fake compact Hermitian symmetric spaces of type other than An (n ≤ 4)." J. Math. Soc. Japan 64 (3) 683 - 731, July, 2012. https://doi.org/10.2969/jmsj/06430683