On Raghunathan’s measure conjecture. (English) Zbl 0763.28012

Ratner, Marina

Ann. Math. (2) 134, No. 3, 545-607 (1991).

Let \(\Gamma\) be a discrete subgroup of a real 2nd countable Lie group with quotient map \(\pi:G\to\Gamma\backslash G\). A probability measure \(\mu\) on \(\Gamma\backslash G\) is called algebraic if there is an \(x\in G\) such that \(\mu(\pi(x)\Lambda(\mu))=1\), where \(\Lambda(\mu)=\{g\in G|\) the action of \(g\) on \(\Gamma\backslash G\) preserves \(\mu\}\). The action of another subgroup \(U\) of \(G\) on \(\Gamma\backslash G\) is measure rigid if every \(U\)-invariant probability measure on \(\Gamma\backslash G\) is algebraic, and \(U\) is measure rigid in \(G\) if its action on \(\Gamma\backslash G\) is measure rigid for every lattice \(\Gamma\subset G\); \(U\) is strictly measure rigid in \(G\) if its action on \(\Gamma\backslash G\) is measure rigid for every discrete subgroup \(\Gamma\subset G\). \(U\) is called unipotent if \(\text{Ad}_ u\) is a unipotent automorphism of the Lie algebra of \(G\) for all \(u\in U\).
Raghunathan’s measure conjecture, as formulated by the author, then states: every unipotent subgroup of a connected Lie group \(G\) is measure rigid. In this paper, the author’s 3rd and last on this topic [cf. also Invest. Math. 101, No. 2, 449-482 (1990; Zbl 0745.28009); Acta Math. 165, No. 3/4, 229-309 (1990; Zbl 0745.28010)], the main theorem is as follows.
Theorem. Every unipotent subgroup of a connected Lie group is strictly measure rigid.

Reviewer: P.Milnes (London / Ontario)

MSC:

28D05	Measure-preserving transformations
22E40	Discrete subgroups of Lie groups
22D40	Ergodic theory on groups
37A99	Ergodic theory

Keywords:

strict measure rigidity; algebraic measure; measure-preserving action; ergodic action; Raghunathan’s measure conjecture; unipotent subgroup of a connected Lie group

Citations:

Zbl 0745.28009; Zbl 0745.28010