Arithmetic fake projective spaces and arithmetic fake grassmannians

We show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at least four such subgroups, which are in fact torsion-free. This, in particular, leads to examples of a fake projective space of dimension 4. Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are also obtained.Comment: 20 pages, the exposition has been improve

Nonexistence of arithmetic fake compact hermitian symmetric spaces of types other than An

We show that there are no arithmetic fake compact hermitian symmetric spaces of type other than An for n>4.Comment: 38 pages, an expanded and improved version of the original article to include consideration of type An for n>4, to appear in Journal of Mathematical Society of Japa