Zariski density in lie groups

A note on subgroups of cofinite volume

“…The oldest of these concerns semisimple Lie groups without compact factors due to Borel (see Theorem 5.5 of [10]). More general density results have been proved in [7] (as well as by others). In [7] a density theorem relevant to this note holds when G is a connected subgroup of GLðn; RÞ, with radical R, G=R has no compact factors and R acts on R n with real eigenvalues.…”

Complete reducibility and Zariski density in linear Lie groups

“…A linear group, G, is called k-minimally almost periodic if each krational representation, ρ of G, with ρ(G) bounded, is trivial. By combining Theorem (2.4), which generalizes the well known density theorem of A. Borel, and Proposition (2.5), both of [6], we get the following:…”

“…The first result deals with arbitrary connected real linear groups and generalizes a result of G.D. Mostow, while the second deals with algebraic groups and their arithmetic subgroups and is parallel to an unpublished result of G. Hochschild. 3 We first recall the density theorem of R. Mosak and the author in [6]. To state it we require some definitions.…”

“…In particular (see [6]) the density theorem holds when G is minimally almost periodic (Furstenberg's case), complex analytic, or is a real analytic group, with radical R, such that G/R has no compact factors and (for example) R acts on V with real eigenvalues. It should also be mentioned that the conclusion of the density theorem need not hold when G/H is compact, but doesn't have finite volume.…”

“…Rather than normality of the subgroup, what is needed here is some finiteness property of H with respect to G. Then follows Corollary 3 which is a generalization, to this setting, of the fact that a continuous representation of a compact group on a Hilbert space is completely reducible. Recalling the density theorem of R. Mosak and the author in [6], we then prove two results which guarantee H is Zariski dense in G. Theorem 4 deals with arbitrary connected real linear groups and their cofinite volume subgroups and generalizes a result of G.D. Mostow, while Theorem 5 characterizes when a Zariski connected algebraic Q-group has the property that its Z-points are Zariski dense. Finally, using Theorem 1, density results and a number of other facts, we prove Theorem 6 which states that for an arbitrary linear algebraic Q-group and a Q-rational representation, complete reducibility of the representation is equivalent to complete reducibility of its restriction to any arithmetic subgroup.…”