On representations of complex hyperbolic lattices

The following superrigidity type theorem for complex hyperbolic lattices is proved in this paper. Let $X=\Gamma \backslash B^n$ be a compact complex ball quotient, $n= 2 $ or $3$. Suppose ${\bf H}^{1,1}$$(X,{\bf C})$ $\cap$ ${\bf H}^2(X,{\bf Z} )$ is generated by the Kähler class of $X$. Then any representation of $\Gamma$ in $GL(n+1,{\bf C})$ can either be deformed to a unitary representation or be extended to a homomorphism from $SU(n,1)$ into $GL(n+1,{\bf C})$.

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Published 1 January 1999