We prove a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups (not necessarily Lie). In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily contains (up to finite index) the lattice.

The approach, as in Margulis’ Normal Subgroup Theorem for lattices, involves, on the one hand, using cohomology to prove a certain group has property (T), and on the other, dynamics of group actions that are strongly not measure-preserving to show amenability.

In particular, the commensurator of any uniform lattice in a simple tree automorphism group is just infinite if and only if the corresponding relative profinite completion is just openly noncompact, partially answering a question of Lubotzky, Mozes and Zimmer.

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 the property (T) “half” of the Margulis-Zimmer conjecture: any subgroup of a lattice in a Lie group which is commensurated by the lattice has relative profinite completion that has property (T).

The main tool is our proof that the natural restriction map of reduced cohomology to a dense subgroup is injective when the ambient group is totally disconnected.

This in turn follows from our main result that every reduced cohomology class contains a unique harmonic representative, allowing us to bring ideas from dynamics into representation theory.

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.