SAT Actions and Rigidity of Lattices
Vanderbilt University
30 Nov 2011

I will present an overview of SAT actions, a type of quasi-invariant group action on a probability space that is the opposite of measure-preserving, and recent work of Y. Shalom and myself on the rigidity of such actions for lattices in the form of our SAT Factor Theorem. I will then explain how this result plays the key role in the previously presented work on Normal Subgroups of Commensurators of Lattices.

Normal Subgroups of Commensurators of Lattices
Vanderbilt University
9 Nov 2011

I will present some results of myself and Y. Shalom. I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.

Normal Subgroups of Commensurators and Rigidity of SAT Actions
Vanderbilt University
26 Aug & 2 Sep & 9 Sep 2011

I will present some results of myself and Y. Shalom in a pair of talks.

During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms.

The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of a potential structure theory for quasi-invariant actions.

The second talk should be understandable even without the background presented in the first though it will be helpful.

Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces
Darren Creutz
Doctoral Dissertation

I will be an Assistant Professor of Mathematics at Vanderbilt University from August 2011 onward.

Awarded the PhD in Pure Mathematics from UCLA awarded 9 June 2011.

Dynamics of SAT Actions
CalTech
2 May 2011

I will present an overview of SAT actions, a class of quasi-invariant actions that are the “opposite” of measure-preserving in a natural way. After presenting key results on SAT and some of my own work (joint with Y. Shalom), I will discuss my new notion of relatively SAT factor maps–the counterpart to relative measure-preserving–and discuss progress toward a structure theory for quasi-invariant actions.

Normal Subgroups of Commensurators and Rigidity of SAT Actions
UCLA
6 & 13 April 2011

I will present an overview of my dissertation research in a pair of talks. During the first talk, I will focus on our Normal Subgroup Theorem for Commensurators of lattices: any normal subgroup of a (dense) commensurator of a lattice in a locally compact group necessarily contains the lattice. Consequences of this theorem will also be discussed: classification of normal subgroups of commensurators; an improved form of Bader-Shalom's normal subgroup theorem for lattices in products; and a partial answer to a question of Lubotzky, Mozes and Zimmer on tree automorphisms. The second talk will focus on our results on group dynamics for quasi-invariant actions that are the main new ingredient required to prove the normal subgroup theorem. I will discuss strongly approximately transitive actions and their various structural and rigidity properties. The talk will conclude with a discussion of our progress on two open questions: the Margulis-Zimmer Conjecture on commensurated subgroups of lattices and a potential structure theory for quasi-invariant actions. The second talk should be understandable even without the background presented in the first. This is joint work with Yehuda Shalom.

Next year I will be a postdoc at Vanderbilt University in Nashville, Tennessee.

Mixing Properties of Random Sequences and “Stochastic Staircase” Transformations
Darren Creutz
(in review)

Quasi-Invariant Group Actions
UCLA Logic Seminar
18 Feb 2011

I will present an overview of the ergodic theory of groups acting quasi-invariantly on probability spaces (meaning the measure is not preserved by the action but the null sets are). Such actions arise naturally in the context of Lie groups acting on symmetric space and automorphisms of trees acting on graphs. The bulk of the talk will be background and introductory material; I will conclude with a description of my own research and results in this area.

Normal Subgroups and Rigidity for Commensurators
Vanderbilt University
28 Feb 2011

We present 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 involves new rigidity theorems for commensurators both in the context of representations and in dynamics, in particular a new factor theorem for SAT actions (the natural opposite of measure-preserving) more general than those for boundaries.
This is joint work with Yehuda Shalom.

Normal Subgroups of Commensurators and SAT Actions
CalTech
20 Jan 2011

I will present a Normal Subgroup Theorem for (dense) commensurators of lattices in arbitrary locally compact groups. In particular, any normal subgroup of a (dense) commensurator of an (integrable) lattice in a simple topological group necessarily virtually contains the lattice.
SAT actions, the natural opposite of measure-preserving, play a key role and we establish several results about them culminating in a Factor Theorem for SAT actions of lattices.
Some consequences of our work, including a new proof of the Normal Subgroup Theorem for lattices in products, will complete my presentation.
Knowledge of commensurators and Normal Subgroup Theorems will not be assumed.
This is joint work with Yehuda Shalom.

Normal Subgroups of (Dense) Commensurators of Lattices
Darren Creutz and Yehuda Shalom
(in preparation)

Harmonic Cocycles and Commensurated Subgroups
Darren Creutz and Yehuda Shalom
(in preparation)

A Normal Subgroup Theorem for Commensurators
Yale University
15 Nov 2010

We present 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 and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable.
This is joint work with Yehuda Shalom.

Mixing, Random Sequences and Rank-One Transformations
Northwestern University
9 Nov 2010

We present new results on "random" sequences (sufficiently general enough to include deterministic sequences such as polynomials) having various mixing- and ergodic-type properties with respect to transformations having certain mixing-type properties. The main application is a proof of mixing on "stochastic staircase" rank-one transformations, a class which includes all previously known examples of mixing rank-one. The talk will consist of a discussion of the mixing- and ergodic-type properties involved, some indications as to the proofs for random sequences, and an introduction to rank-one transformations with an indication of how one proves mixing.

Normal Subgroup and Factor Theorems for Commensurators
University of Illinois: Chicago
8 Nov 2010

We present 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, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable. We will focus more on the amenability half of the proof, in particular our new ”Factor Theorem” which facilitates the proof (and which is of independent interest). This is join work with Yehuda Shalom.

A Normal Subgroup Theorem for Commensurators of Lattices
AMS Western Meeting
9 Oct 2010

We prove a statement akin to Margulis’ Normal Subgroup Theorem for lattices in Lie groups, but our Theorem applies not to lattices but to commensurators of lattices. We show that any infinite normal subgroup of a (dense) commensurator of a lattice in a Lie group necessarily intersects the lattice in a finite index subgroup. We then develop this into a correspondence between normal subgroups of the commensurator and open normal subgroups of the relative profinite completion.
The approach, as in Margulis’ Theorem, involves, on the one hand, using cohomology and rigidity theory to prove a certain group has property (T), and on the other hand, Furstenberg’s Boundary Theory to prove this group is also amenable. We will focus more on the amenability half of the proof, in particular our new ”Factor Theorem” which facilitates the proof (and which is of independent interest).

Mixing on Rank-One Transformations
Darren Creutz and Cesar Silva
Studia Mathematica

Superstability and Finite Time Extinction for C0-Semigroups
D. Creutz, M. Mazo Jr. and C. Preda
(in review)

Robert Sorgenfrey Distinguished Teaching Award
UCLA
2010

Superstability and Finite-Time Extinction for Semigroups
UCLA
27 Apr 2010

VIGRE Instructorship
UCLA
2010

VIGRE Fellowship
UCLA
2005-2009

Poisson Boundaries and Their Applications
UCLA
Jan 2009

Rank-One Actions, Mixing and Singular Spectra
UCLA
Mar 2007

SMALL Research Internship
Williams College
2001-2004

Mixing on a Class of Rank-One Transformations
Darren Creutz and Cesar Silva
J. Ergodic Th. & Dyn. Sys.

Rank-One Mixing and Dynamical Averaging
Darren Creutz
Honors Thesis