Algebraic hulls and the Følner property (Q1924203)

A special case of the main results asserts that the semisimple component of the algebraic hull of a cocycle for the action of a simple noncompact Lie group \(G\) with finite center on a space with finite \(G\)-invariant measure coincides with the algebraic hull of the restriction of the action to a co-Følner subgroup \(\Lambda\) (that is a Zariski dense discrete subgroup \(\Lambda\subset G\) such that there exists a \(G\)-invariant measure on \(G/\Lambda\)). As a consequence, we have that a homomorphism of \(G\) with semisimple Zariski closure has the same Zariski closure when restricted to \(\Lambda\). In fact, the theorems are proven in much greater generality and examples are given to demonstrate the sharpness of most of the hypotheses.