Arithmetic fake compact Hermitian symmetric spaces of type 𝐴₃

抄録

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. Arithmetic fake compact Hermitian symmetric spaces of types other than 𝐴₁ and 𝐴₃ have been classified in our earlier work and the work of Cartwright–Steger. There are many known examples of type 𝐴₁ and there may not be a convenient way to describe them all, but there are already descriptions under some restrictions given in Shavel and Linowitz–Stover–Voight. In this article, we study the remaining type 𝐴₃. We show that there are possibly a handful of explicit candidates.