Character Rigidity for Lattices and Commensurators
Darren Creutz and Jesse Peterson
We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure preserving actions, and also answers a question of Connes for such groups.
Stabilizers of Actions of Lattices in Products of Groups
Ergodic Theory and Dynamical Systems
We prove that any ergodic nonatomic probability-preserving action of an irreducible lattice in a semisimple group, at least one factor being connected and higher-rank, is essentially free. This generalizes the result of Stuck and Zimmer that the same statement holds when the ambient group is a semisimple real Lie group and every simple factor is higher-rank.
We also prove a generalization of a result of Bader and Shalom 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.
Stabilizers of Ergodic Actions of Lattices and Commensurators
Darren Creutz and Jesse Petersoncite
Transactions of the AMS (Vol. 369 p. 4119–4166)
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.
Contractive Spaces and Relatively Contractive Maps
AMS Contemporary Mathematics (Vol. 678 p. 103–132)
We present an exposition of contractive spaces and of relatively contractive maps. Contractive spaces are the natural opposite of measure-preserving actions and relatively contractive maps the natural opposite of relatively measure-preserving maps. These concepts play a central role in the work of the author and J.~Peterson on the rigidity of actions of semisimple groups and their lattices and have also appeared in recent work of various other authors. We present detailed definitions and explore the relationship of these phenomena with other aspects of the ergodic theory of group actions, proving along the way several new results, with an eye towards explaining how contractiveness is intimately connected with rigidity phenomena.
A Normal Subgroup Theorem for Commensurators of Lattices
Darren Creutz and Yehuda Shalomcite
Groups, Geometry and Dynamics (Vol. 8 p. 789–810)
We establish a general normal subgroup theorem for commensurators of lattices in locally compact groups. While the statement is completely elementary, its proof, which rests on the original strategy of Margulis in the case of higher rank lattices, relies heavily on analytic tools pertaining to amenability and Kazhdan's property (T). It is a counterpart to the normal subgroup theorem for irreducible lattices of Bader and the second named author, and may also be used to sharpen that result when one of the ambient factors is totally disconnected.
Mixing on Stochastic Staircase Transformations
We prove mixing on a general class of rank-one transformations containing all known examples of rank-one mixing, including staircase transformations and Ornstein's constructions, and a variety of new constructions.
Mixing on Rank-One Transformations
Darren Creutz and Cesar Silvacite
Studia Mathematica (Vol. 199 p. 43–72)
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.
Mixing on a Class of Rank-One Transformations
Darren Creutz and Cesar Silvacite
Ergodic Theory and Dynamical Systems (Vol. 24 p. 407–440)
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.
Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces
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.
Rank-One Mixing and Dynamical Averaging
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.