We present a new proof of the result of Stuck and Zimmer [SZ94] that any ergodic probability-preserving action of an irreducible lattice in a semisimple real Lie group, each simple factor of higher-rank, is essentially free, and generalize to the case when only one simple factor is of higher-rank.

We also prove a generalization of a result of Bader and Shalom [BS06] by showing that any probability-preserving action of a product of simple groups, at least one with property $(T)$, which is ergodic for each simple subgroup is either essentially free or essentially transitive.

Our method involves the study of relatively contractive maps and the Howe-Moore property, rather than the relaying on algebraic properties of semisimple groups and Poisson boundaries, and introduces a generalization of the ergodic decomposition to invariant random subgroups of independent interest.

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices.

The above are derived from more general results on groups with the Howe-Moore property and property (T). We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.

Let G be a compactly generated locally compact second countable group that is not a compact extension of an abelian group and let L be a finitely generated integrable lattice in G. Assume that the commensurator of L is dense in $G$ and has finite intersection with every closed normal noncocompact subgroup of G.

Then any infinite normal subgroup of the commensurator contains L up to finite index. Moreover, there is a one-one, onto correspondence between commensurability classes of infinite normal subgroups of the commensurator and of open normal subgroups of the relative profinite completion.

As a consequence, an irreducible lattice in G x H, a product of compactly generated locally compact second countable groups, G nondiscrete and H totally disconnected, is just infinite if and only if G is just noncompact and H has no infinite index open normal subgroups.

Margulis’ Normal Subgroup and Arithmeticity Theorems classify completely the lattices in higher-rank algebraic groups as being (up to finite index) the simple groups consisting of integer points of algebraic groups. We generalize the Normal Subgroup Theorem to commensurators of lattices in arbitrary locally compact groups and in particular obtain that the commensurator of a uniform tree lattice in a regular tree is simple (up to finite index) if and only if a corresponding completion is. We also make progress on the Margulis-Zimmer Conjecture: a generalization of the Arithmeticity Theorem to commensurated--geometrically normal--subgroups by establishing the property (T) “half” of the conjecture.

In keeping with the methods of Margulis’ work and of rigidity theory in general, we prove new results in the area of the dynamics of group actions with quasi-invariant measures and in the area of unitary representation theory that in turn lead to our structural results on commensurators of lattices in arbitrary groups and commensurated subgroups of lattices in algebraic groups.

The dynamics of group actions with quasi-invariant measures–the study of group actions on measure algebras preserving the ideal of null sets, but not necessarily the measure itself–is the natural setting for dynamics of nonamenable groups (such as those we are concerned with). We develop some foundational results on strongly approximately transitive actions–the dynamical opposite of measure-preserving–and prove a “SAT Factor Theorem” along the lines of Margulis’ original boundary-based factor theorem that plays a key role in the proof of our Normal Subgroup Theorem.

In the realm of unitary representations we prove an existence and uniqueness result for harmonic cocycles–namely, in any reduced cohomology class there exists a unique harmonic representative. Consequences of this include our progress on the Margulis-Zimmer Conjecture.

We prove mixing on a general class of rank-one transformations, termed “stochastic staircases”, containing all known examples of rank-one mixing, including staircase transformations, Ornstein’s constructions and a new class of “random staircases” (spacers chosen uniformly at random with positive density). This is a consequence of our result that a generalized class of random sequences (those with iid increments) are universally ergodic (and universally totally ergodic, a notion we introduce).

We prove mixing on rank-one transformations is equivalent to “the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums”. In particular, all staircase transformations (and polynomial staircase transformations) are mixing, answering Adams' question.

A new approach to superstability and finite time extinction of strongly continuous semigroups is presented, unifying known results and providing new criteria for these conditions to hold analogous to the well-known Pazy condition for stability. That finite time extinction implies superstability which is in turn equivalent to several (both known and new) conditions follow from this new approach in a consistent fashion. Examples that the converse statements fail are constructed, in particular, an answer to a question of Balakrishnan on superstable systems not exhibiting finite time extinction.

We prove that a rank-one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences.

The application of our theorem shows that the class of polynomial rank-one transformations, rank-one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations implying in particular Adams’ result on staircase transformations. Another application yields a new proof that Ornstein’s class of rank-one transformations constructed using “random spacers” are almost surely mixing transformations.

Addenda and Errata

A rank-one transformation is defined by a sequence of positive integers, the sequence of cuts, and a dynamical sequence of nonnegative integers, the sequence of spacers, that are used to repeatedly cut and stack a single column. Our main result is that rank-one transformations satisfying a condition called restricted growth and such that the spacer sequence is uniformly ergodic with respect to the transformation are mixing transformations.

The restricted growth condition limits the total variation in the spacer sequence and is a generalization of a condition, equivalent to restricted growth for staircase transformations, given by Adams, that is sufficient for staircase transformations to be mixing, while the uniform ergodicity of the spacer sequence is a generalization of the notion of uniform Cesaro transformations used by Adams to show mixing on staircases.

The application of our concepts and results to a class of rank-one transformations, a class we call generalized staircase transformations, yields a variety of rank-one mixing transformations with explicit constructions.