### Talk: Relativized Furstenberg Entropy and an In... Mathematics

**Relativized Furstenberg Entropy and an Information Theory of Joinings**

U.S. Naval Academy

*16 Oct 2017*

The Furstenberg entropy of a nonsingular action of a group on a probability space (a G-space) is a numerical measure of how far the action is from being measure-preserving. This entropy has found many applications due to its close connection to the Poisson boundary; in particular, Kaimanavich and Vershik proved a Shannon-McMillan-Breiman theorem for actions centered on this concept. In this talk, I will present a generalization of the Furstenberg to entropy to maps between G-spaces. This generalization enjoys all of the properties one would expect from an entropy and forms the basis for an information-theoretic approach to joinings of G-spaces. In particular, I introduce concepts of mutual and conditional information which satisfy inequalities corresponding to their classical counterparts. Some applications to groups with property (T) will be discussed (and some conjectures), time permitting.

### Talk: Set Theory Past ZFC: Large Cardinals, Forcing, and Indep... Mathematics

**Set Theory Past ZFC: Large Cardinals, Forcing, and Independence**

U.S. Naval Academy

*3 April 2017*

Most everyone "knows" that math is built on the foundation of Zermelo-Frankl (ZF) set theory, but most of us don't know about the axioms that go beyond ZF(C). Godel's proof that the Axiom of Choice is consistent with ZF, followed by Cohen's proof that its negation is also consistent with ZF, was the first indication that there is more going on than was first thought. The same result (by the same people) showing that the Continuum Hypothesis is independent of ZFC led to a foundational crisis in mathematical logic. I will give a high-level overview of these results and then explain the various additional axioms that have been proposed (many of which are now taken as "standard" by set theorists) to resolve these issues. Time permitting (or in a follow-up talk) I will discuss how this relates to Godel's Incompleteness Theorem and model theory.

### Talk: The Normal Subgroup Theorem for Lattices in Products (Pr... Mathematics

**The Normal Subgroup Theorem for Lattices in Products (Property (T))**

U.S. Naval Academy

*6 Feb 2017*

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads into my work on a conjecture of Margulis and Zimmer about the nature of commensurated subgroups of lattices. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). The first discussed the amenability proof; this talk will present the (T) half. In particular, this talk will be a standalone talk and will not assume knowledge of what was talked about in part one.

### Talk: The Normal Subgroup Theorem for Lattices in Products (Am... Mathematics

**The Normal Subgroup Theorem for Lattices in Products (Amenability)**

U.S. Naval Academy

*23 Jan 2017*

Margulis' Normal Subgroup Theorem states that if Gamma is an irreducible lattice in a higher-rank semisimple Lie group with trivial center then every nontrivial normal subgroup of Gamma has finite index. Moving away from Lie groups, Bader and Shalom proved that the same result holds for lattices in products of arbitrary simple nondiscrete locally compact groups. I will present joint work with Y. Shalom which gives a new proof of this result and explain how it leads to my work with J. Peterson that every ergodic action of an irreducible lattice in a product of higher-rank semisimple groups on a nonatomic probability space is essentially free. The proof is in two distinct halves (as was Margulis'): we prove Gamma / N is finite by showing it is both amenable and has Kazhdan's property (T). Part one will discuss the amenability proof; part two (to be scheduled) will discuss (T). Note: part two will not rely on part one; each talk will be stand-alone.

### Talk: Random Walks and Harmonic Functions on Groups Mathematics

**Random Walks and Harmonic Functions on Groups**

U.S. Naval Academy

*7 November 2016*

A natural question in geometric group theory is to study the random walk of a finitely generated group. Specifically, for a probability distribution mu on a finite generating set S, one considers the behavior of the random walk on the Cayley graph built from S with law mu (meaning at each step in the walk, we choose which edge in S to follow according to mu). In particular, one considers the exit boundary of the walk--the space of all distinct paths to infinity. Another natural question is to study the space of bounded mu-harmonic functions on G: functions f : G --> Reals such that for each g in G, Sum_{s in S} f(gs) mu(s) = f(g). The classical Dirichlet problem establishes a correspondence between bounded harmonic functions on SL_2 (the fractional linear transformations) and bounded measurable functions on the unit circle. I will present Furstenberg's Poisson Boundary construction which establishes that random walks on groups and harmonic functions are both determined by the bounded measurable functions on the ``boundary" of the random walk. In particular, the bounded harmonic functions are in one-one correspondence with the bounded measurable functions on the boundary. A concrete example of this is the free nonabelian group on two generators F_2: the Cayley graph (for the usual generating set) is the regular 4-tree and the natural weighing is to give all 4 directions equal weight; the boundary here is the ``big circle", the boundary of the 4-tree, and the harmonic functions on F_2 are in one-one correspondence with L^infinity of the big circle.

### Talk: Ergodic Actions of Lattices in Higher-Rank Semisimple Gr... Mathematics

**Ergodic Actions of Lattices in Higher-Rank Semisimple Groups**

University of Maryland

*6 October 2016*

Lattices in higher-rank semisimple groups arise naturally in many areas of mathematics, and include groups such as SL_n[Z] for n >= 3. These groups exhibit a variety of rigidity properties, most notably the results of Margulis--the Normal Subgroup Theorem that every nontrivial normal subgroup of an irreducible lattice in a center-free higher-rank semisimple group has finite index and the Superrigidity Theorem that every isomorphism of such a lattice into any algebraic group either has precompact image or extends to the ambient semisimple group. I will present work of myself and J. Peterson generalizing both of these theorems. The main focus of the talk will be on our theorem that every ergodic action of such a lattice on a nonatomic probability space is essentially free (taking the action to be the Bernoulli shift on the lattice modulo a normal subgroup recovers the NST); the proof of which involves a careful understanding of the dynamics of the Poisson boundary and of Howe-Moore groups. I will also present (largely without proof) our operator-algebraic superrigidity theorem that any representation of such a lattice as unitary operators on a finite von Neumann algebra is either finite-dimensional (hence coming from a quotient by a finite index normal subgroup) or extends to the entire group von Neumann algebra of the lattice.

### Talk: Mixing and Rank-One Transformations Mathematics

**Mixing and Rank-One Transformations**

U.S. Naval Academy

*3 October 2016*

Ergodic theory (the classical theory) is the study of transformations on probability spaces. This talk will introduce the basic notions of the theory: ergodicity and various forms of mixing; then introduce a class of transformations constructed by an intuitive process of "cutting and stacking". These transformations (rank-one transformations) have been studied since the 1940s as a means to understand the mixing notions. The talk will present some of the main results beginning with Chacon's proof of weak mixing not implying strong mixing and Ornstein's proof of the existence of zero entropy transformations with no square root which are strong mixing and conclude with the presenter's work (partly joint with C. Silva) on constructing explicit examples of such transformations.

### Talk: Character Rigidity for Lattices in Lie Groups Mathematics

**Character Rigidity for Lattices in Lie Groups**

U.S. Naval Academy

*26 August 2016*

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace); the classical theory of characters is the first step in the classification of finite simple groups and culminates in the Peter-Weyl theorem for compact groups. I will present the results of J. Peterson and myself that the only characters on lattices in semisimple groups are the left-regular character and the classical characters. This is in actuality operator-algebraic superrigidity for lattices, answering a question of Connes. The main idea is to bring dynamics into the operator-algebraic picture; the second half of the talk will focus on the ergodic-theoretic ideas of contractiveness and the Poisson boundary and how these ideas lead to operator-algebraic results.

### Talk: Rigidty Theory of Lattices in Semisimple Groups Mathematics

**Rigidty Theory of Lattices in Semisimple Groups**

U.S. Naval Academy

*3 Feb 2016*

### Talk: The Information Theory of Joinings Mathematics

**The Information Theory of Joinings**

Vanderbilt University

*22 January 2016*

I will present ongoing research into an area I am developing based on the idea of treating joinings of quasi-invariant actions of groups on probability spaces along similar lines are treating random variables as representing information, in particular I consider the ``mutual information" of two spaces in terms of their joinings. Furstenberg entropy is a numerical measure of how far a quasi-invariant action of a group on a probability space is from measure-preserving. The main new tool I introduce is a relative version of this entropy measuring how far a homomorphism between such spaces is from being relatively measure-preserving. I show that it enjoys the properties one would expect such as additivity over compositions and apply this notion to develop an “information theory” of joinings proving analogues of many of the key theorems in the information theory of random variables.

### Publication: Stabilizers of Actions of Lattices in Products of... Mathematics

**Stabilizers of Actions of Lattices in Products of Groups**

Darren Creutz

*Ergodic Theory and Dynamical Systems*

### Publication: Contractive Spaces and Relatively Contractive Maps Mathematics

**Contractive Spaces and Relatively Contractive Maps**

Darren Creutz

*AMS Contemporary Mathematics*

### Talk: Co-Organizer, Special Session: Classification Problems i... Mathematics

**Co-Organizer, Special Session: Classification Problems in Operator Algebras**

AMS Joint Mathematics Meetings

*11 January 2015*

### Talk: Harmonic Maps on Groups and Property (T) Mathematics

**Harmonic Maps on Groups and Property (T)**

Noncommutative Geometry and Operator Algebras Spring Institute

*6 May 2015*

Furstenberg's boundary theory allows us to characterize amenability in terms of the absence of bounded harmonic functions on the group. Building on joint work with Y. Shalom, I will present a similar method for characterizing Kazhdan's Property (T) in terms of the absence of certain harmonic maps on the group. Together, these results give some insight into a potential unified proof of Margulis' Normal Subgroup Theorem (and other Normal Subgroup Theorems).

### Award: Co-Organizer, AMS Special Session on Classification Pro... Mathematics

**Co-Organizer, AMS Special Session on Classification Problems in Operator Algebras**

Joint Mathematics Meetings

*2015*

### Publication: A Normal Subgroup Theorem for Commensurators of L... Mathematics

**A Normal Subgroup Theorem for Commensurators of Lattices**

Darren Creutz and Yehuda Shalom

*Groups, Geometry and Dynamics*

### Talk: Operator Algebraic Superrigidity for Lattices and Commen... Mathematics

**Operator Algebraic Superrigidity for Lattices and Commensurators**

Northwestern University

*3 Dec 2013*

### Talk: Rigidity for Characters on Lattices and Commensurators Mathematics

**Rigidity for Characters on Lattices and Commensurators**

Vanderbilt University

*30 Oct 2013*

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, answering a question of Connes and generalizing our result that every nonatomic probability-preserving action of such a group is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it into the operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

### Publication: Character Rigidity for Lattices and Commensurators Mathematics

**Character Rigidity for Lattices and Commensurators**

Darren Creutz and Jesse Peterson

*(in review)*

### Talk: Operator-Algebraic Superrigidity for Lattices Mathematics

**Operator-Algebraic Superrigidity for Lattices**

AMS Special Session on Classification Problems in Operator Algebras, Baltimore Maryland

*15 Jan 2014*

I will present an overview of my recent work, both joint with J. Peterson and solo, classifying the possible actions of lattices in semisimple groups, and more generally, products of groups with the Howe-Moore property. The main result is that, provided at least one simple factor in the ambient group has property (T) (is of higher-rank), every ergodic probability-preserving action of such a lattice on a nonatomic space is essentially free. I will also explain more recent work, joint with J. Peterson, on the rigidity for characters on such lattices, the noncommutative analogue of the statement on actions.

### Talk: Character Rigidity for Lattices and Commensurators Mathematics

**Character Rigidity for Lattices and Commensurators**

Vanderbilt University

*27 Sep 2013*

Characters on groups (positive definite conjugation-invariant functions) arise naturally both from probability-preserving actions (the measure of the set of fixed points) and unitary representations on finite factors (the trace). I will present joint work with J. Peterson showing the nonexistence of nontrivial characters for irreducible lattices in semisimple groups and for their commensurators. Consequently, any finite factor representation of such a group generates either the left regular representation or a finite-dimensional representation, generalizing our earlier result that every nonatomic probability-preserving action of such groups is essentially free. The key new idea is to use the contractive nature of the Poisson boundary to bring it in operator algebraic setting and along with it the rigidity behavior of lattices in their ambient groups.

### Publication: Mixing on Stochastic Staircase Transformations Mathematics

**Mixing on Stochastic Staircase Transformations**

Darren Creutz

*(in review)*

### Talk: Stabilizers of Actions of Groups and Invariant Random Su... Mathematics

**Stabilizers of Actions of Groups and Invariant Random Subgroups**

Vanderbilt University

*26 Apr 2013*

As an introduction to the upcoming Shanks workshop on von Neumann Algebras and Ergodic Theory, I will introduce the basic notions involved with invariant random subgroups. Actions of groups give rise to invariant random subgroups via the stabilizer map; I will show how to construct an action that gives a prescribed invariant random subgroup as its stabilizers. Then I will discuss notions such as subgroups of random subgroups (due to myself and J. Peterson) and quotienting out by random subgroups.

### Talk: Stabilizers of Ergodic Actions of Product Groups and Lat... Mathematics

**Stabilizers of Ergodic Actions of Product Groups and Lattices in Products**

Shanks Workshop on von Neumann Algebras and Ergodic Theory, Vanderbilt University

*28 Apr 2013*

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of a semisimple real Lie group with every simple factor of higher-rank is essentially free and likewise for lattices in such groups. Bader and Shalom, following a different approach, showed that any properly ergodic action of a product of two simple groups with property (T) is essentially free, but their methods do not yield information about lattices.

I will present recent work expanding on the approach of Bader and Shalom generalizing the results of Stuck and Zimmer and of Bader and Shalom to the case when only one factor has (T) and obtaining a classification statement for actions of lattices in products of simple Howe-Moore groups.

### Talk: Stabilizers of Actions of Product Groups and Lattices in... Mathematics

**Stabilizers of Actions of Product Groups and Lattices in Product Groups**

Vanderbilt University

*5 April 2013*

I will present my recent work on the stabilizers of actions of products of groups and irreducible lattices in products. The main results are a classification of all possible stabilizer groups for actions of products of Howe-Moore groups, at least one of which has (T), and a classification statement for actions of lattices in such products. In contrast to previous work (joint with J. Peterson) on stabilizers, the approach taken here does not involve writing lattices as commensurators and therefore applies even in the case when neither of the ambient groups are totally disconnected and in this sense complement the previous work.

### Talk: Mixing on Rank-One Transformations Mathematics

**Mixing on Rank-One Transformations**

Vanderbilt University

*25 Jan 2013*

In this talk on a more classical part of ergodic theory, that of Z-actions, I will explain the construction of rank-one transformations via cutting and stacking that goes back to von Neumann and Kakutani and has been used to create examples and counterexamples of various mixing-like properties. Following the explanation of the subject, I will present some of my work on when such transformations are mixing. Some of the results presented are joint work with Cesar Silva.

### Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

**Stabilizers of Ergodic Actions of Lattices and Commensurators**

University of California: San Diego

*16 Nov 2012*

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free.

I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of PSL_{n}(Q), for n ≥ 3, is essentially free.

The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive actions.

### Talk: Poisson Boundaries, Harmonic Functions and Random Walks ... Mathematics

**Poisson Boundaries, Harmonic Functions and Random Walks on Groups**

Vanderbilt University

*9 Nov & 5 Dec 2012*

I will present the construction of the Poisson Boundary of a group, originally defined by Furstenberg, and explain its various properties and applications. The Poisson Boundary can be thought of as the exit boundary of a random walk on the group and can be identified with the space of harmonic functions on the group. The first talk will focus on the construction of the Poisson Boundary and various results due primarily to Furstenberg and Zimmer about boundaries. The second talk will focus on the dynamical behavior of the boundary and its applications to ergodic theory.

### Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

**Stabilizers of Ergodic Actions of Lattices and Commensurators**

Vanderbilt University

*19 Sep 2012*

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Normal Subgroup Theorem for Commensurators of Lattices, due to the first author and Shalom, to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In the second talk, I will explain how the results developed in the first talk lead to the conclusions about S-arithmetic lattices and to lattices in products. The main ideas involve using the Howe-Moore property and property (T) to ensure that actions of the ambient groups satisfy the necessary conditions. Another key idea in studying lattices in products is that most lattices in product groups are isomorphic to the commensurator of a lattice in one of the component groups.

### Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

**Stabilizers of Ergodic Actions of Lattices and Commensurators**

Williams College Ergodic Theory Conference

*28 July 2012*

The Margulis Normal Subgroup Theorem states that any normal subgroup of an irreducible lattice in a center-free higher-rank semisimple Lie group is of finite index. Stuck and Zimmer, expanding on Margulis' approach, showed that any properly ergodic probability-preserving ergodic action of such a lattice is essentially free. I will present similar results: my work with Y. Shalom on normal subgroups of lattices in products of simple locally compact groups and normal subgroups of commensurators of lattices, and my work with J. Peterson generalizing this result to stabilizers of ergodic probability-preserving actions of such groups. As a consequence, S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) as do lattices in certain product groups. In particular, any nontrivial ergodic probability-preserving action of PSL

_{n}(Q), for n at least 3, is essentially free. The key idea in the study of normal subgroups is considering nonsingular actions which are the extreme opposite of measure-preserving. Somewhat surprisingly, the key idea in understanding stabilizers of probability-preserving actions also involves studying such actions and the bulk of our work is directed towards properties of these contractive, or SAT, actions.

### Publication: Stabilizers of Ergodic Actions of Lattices and Co... Mathematics

**Stabilizers of Ergodic Actions of Lattices and Commensurators**

Darren Creutz and Jesse Peterson

*Transactions of the AMS*

### Talk: Stabilizers of Ergodic Actions of Lattices and Commensur... Mathematics

**Stabilizers of Ergodic Actions of Lattices and Commensurators**

UCLA Workshop on von Neumann Algebras and Ergodic Theory

*26 May 2012*

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of PSLn(Q), for n at least three, is essentially free. This is joint work with Jesse Peterson.

### Talk: Stabilizers for Ergodic Actions of Commensurators Mathematics

**Stabilizers for Ergodic Actions of Commensurators**

Vanderbilt University

*6 April 2012*

A strong generalization of the Margulis Normal Subgroup Theorem, due to Stuck and Zimmer, states that any properly ergodic finite measure-preserving action of an irreducible lattice in a center-free semisimple Lie group with all simple factors of higher-rank is essentially free. We present a similar result generalizing the Creutz-Shalom Normal Subgroup Theorem for Commensurators of Lattices to actions of commensurators. As a consequence, we show that S-arithmetic lattices enjoy the same properties as the arithmetic lattices (the Stuck-Zimmer result) and that lattices in certain product groups do as well. In particular, any nontrivial ergodic measure-preserving action of $\mathrm{PSL}_{n}(\mathbb{Q})$, for $n \geq 3$, is essentially free.

### Talk: The Property (T) "Half" of the Margulis-Zimmer Conjectur... Mathematics

**The Property (T) “Half” of the Margulis-Zimmer Conjecture**

Vanderbilt University

*29 Feb 2012*

Generalizing the Margulis Normal Subgroup Theorem, Margulis and Zimmer conjectured that any subgroup of a lattice in a higher-rank Lie group which is commensurated by the lattice is (up to finite index) of a standard form. I will present some of my work on property (T) for totally disconnected groups and countable dense subgroups and explain how it provides "half" of the solution to the conjecture. This is joint work with Yehuda Shalom.

### Talk: Property (T) for Certain Totally Disconnected Groups Rel... Mathematics

**Property (T) for Certain Totally Disconnected Groups Related to a Conjecture of Margulis and Zimmer**

Vanderbilt University

*17 Feb 2012*

I will present some of my work on reduced cohomology and property (T) for totally disconnected groups and dense countable subgroups. The primary application of this work is to show property (T) for a class of totally disconnected groups arising from a conjecture of Margulis and Zimmer regarding the classification of all commensurated subgroups of lattices in higher-rank Lie groups. The key idea in our work is to expand on Kleiner's work on the energy of a cocycle (the idea of which goes back to Mok) and derive a very general result about energy and reduced cohomology. This is joint work with Yehuda Shalom.

### Talk: SAT Actions and Rigidity of Lattices Mathematics

**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.

### Talk: Normal Subgroups of Commensurators of Lattices Mathematics

**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.

### Talk: Normal Subgroups of Commensurators and Rigidity of SAT A... Mathematics

**Normal Subgroups of Commensurators and Rigidity of SAT Actions**

Vanderbilt University

*26 Aug & 2 & 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.

### Publication: Commensurated Subgroups and the Dynamics of Group... Mathematics

**Commensurated Subgroups and the Dynamics of Group Actions on Quasi-Invariant Measure Spaces**

Darren Creutz

*Doctoral Dissertation*

### Assistant Professor at Vanderbilt Mathematics | Teaching

### Doctor of Philosophy Mathematics

### Talk: Dynamics of SAT Actions Mathematics

**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.

### Talk: Normal Subgroups of Commensurators and Rigidity of SAT A... Mathematics

**Normal Subgroups of Commensurators and Rigidity of SAT Actions**

University of California: Los Angeles

*6 Apr & 13 Apr 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.

### Postdoctoral Position at Vanderbilt Mathematics

### Talk: Quasi-Invariant Group Actions Mathematics

**Quasi-Invariant Group Actions**

University of California: Los Angeles

*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.

### Talk: Normal Subgroups and Rigidity for Commensurators Mathematics

**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.

### Talk: Normal Subgroups of Commensurators and SAT Actions Mathematics

**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.

### Talk: A Normal Subgroup Theorem for Commensurators Mathematics

**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.

### Talk: Mixing, Random Sequences and Rank-One Transformations Mathematics

**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.

### Talk: Normal Subgroup and Factor Theorems for Commensurators Mathematics

**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.

### Talk: A Normal Subgroup Theorem for Commensurators of Lattices Mathematics

**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).

### Publication: Mixing on Rank-One Transformations Mathematics

**Mixing on Rank-One Transformations**

Darren Creutz and Cesar Silva

*Studia Mathematica*

### Award: Robert Sorgenfrey Distinguished Teaching Award Mathematics

**Robert Sorgenfrey Distinguished Teaching Award**

University of California: Los Angeles

*2010*

### Talk: Superstability and Finite-Time Extinction for Semigroups Mathematics

**Superstability and Finite-Time Extinction for Semigroups**

University of California: Los Angeles

*27 Apr 2010*

### Award: VIGRE Instructorship Mathematics

**VIGRE Instructorship**

University of California: Los Angeles

*2010*

### Award: VIGRE Fellowship Mathematics

**VIGRE Fellowship**

University of California: Los Angeles

*2005-2009*

### Talk: Poisson Boundaries and Their Applications Mathematics

**Poisson Boundaries and Their Applications**

University of California: Los Angeles

*Jan 2009*

### Talk: Rank-One Actions, Mixing and Singular Spectra Mathematics

**Rank-One Actions, Mixing and Singular Spectra**

University of California: Los Angeles

*Mar 2007*

### Award: SMALL Research Internship Mathematics

**SMALL Research Internship**

Williams College

*2001-2004*

### Publication: Mixing on a Class of Rank-One Transformations Mathematics

**Mixing on a Class of Rank-One Transformations**

Darren Creutz and Cesar Silva

*Ergodic Theory and Dynamical Systems*

### Publication: Rank-One Mixing and Dynamical Averaging Mathematics

**Rank-One Mixing and Dynamical Averaging**

Darren Creutz

*Honors Thesis*