Nonarithmetic superrigid groups: Counterexamples to Platonov's conjecture

Hyman Bass and Alexander Lubotzky

Review from Zentralblatt MATH:

A finitely generated group $\Gamma$ is called representation superrigid if for all its finite dimensional complex representations $\rho$ the dimension of the Zariski closure of $\rho(\Gamma)$ remains bounded. This property implies that $\Gamma$ is representation rigid, i.e., that it admits only finitely many non-isomorphic irreducible representations of any given dimension. The main result of the paper is as follows. Let $\Gamma$ be a cocompact lattice in the real rank 1 group $G$ of type $F_4$. Then there exist a finite index normal subgroup $\Gamma_1$ of $\Gamma$ and an infinite index subgroup $\Lambda$ of $\Gamma_1\times\Gamma_1$, containing the diagonal, such that any representation of $\Lambda$ extends uniquely to a representation of $\Gamma_1\times\Gamma_1$; the groups $\Lambda$ and $\Gamma_1\times\Gamma_1$ are representation superrigid, and all their representations are semi-simple; $\Lambda$ is not isomorphic to a lattice in any product of groups $H(k)$, where $H$ is a linear algebraic group over a local field $k$. The proof relies on two opposite properties of $\Gamma$: it is superrigid in $G$ and hyperbolic in the sense of Gromov. The result contradicts a conjecture of Platonov claiming that a rigid linear group must be of arithmetic type.

Reviewed by A.L.Onishchik

Keywords: representation; group of arithmetic type; superrigid group; lattice

Classification (MSC2000): 22E40

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