On the Betti Numbers of Compact Rank 2 Locally Symmetric Spaces

Abstract

We obtain upper bounds for the second Betti numbers of compact rank 2 locally symmetric spaces, namely $\Gamma\backslash SL(3)/SO(3)$, $\Gamma\backslash Sp(4)/U(2)$, and $\Gamma\backslash G_{2(2)}/SO(4)$, where $\Gamma$ is a cocompact, torsion free lattice. We use representation theory and directly apply the techniques of Di Cerbo and Stern in \cite{dicerbo2019price}. In the case of $\Gamma\backslash Sp(4)/U(2)$, we also use unitary holonomy, the complex structure operator that arises from it and the (p,q) decomposition of exterior powers to obtain stronger bounds. In particular, the bounds we provide on the Betti numbers of $\Gamma\backslash Sp(4)/U(2)$ and $\Gamma\backslash G_{2(2)}/SO(4)$ are exponential bounds involving injectivity radius. However, the bound we obtained for $\Gamma\backslash SL(3)/SO(3)$ is a weaker polynomial one.