Brandon SewardCourant Instructor / Assistant Professorbseward@cims.nyu.edu Courant Institute of Mathematical Sciences Warren Weaver Hall, Office 1122251 Mercer St.New York, New York10012 Curriculum Vitae Brief Bio     I received my PhD from the University of Michigan in 2015 under the supervision of Ralf Spatzier. During the 2015/2016 academic year I was a postdoc at the Hebrew University of Jerusalem working with Mike Hochman. I started at the Courant Institute in the fall of 2016 and have been working with Tim Austin. Research      My research is spread between the areas of ergodic theory, topological dynamics, descriptive set theory, and group theory. The underlying theme uniting my work is that I study geometric and combinatorial phenomena (involving graphs, trees, colorings, matchings, families of independent random variables, etc.) either on groups themselves or, more often, on the orbits of their actions, subject to measure-theoretic, topological, or Borel-measurability constraints. My past research has focused on entropy theory for probability measure preserving actions of countable non-amenable groups, on the theory of hyperfinite Borel equivalence relations, on descriptive graph combinatorics, on the dynamics of topological Bernoulli shifts, and on geometric and combinatorial group theory.
 Publications and Preprints 23.    Bernoulli shifts with bases of equal entropy are isomorphic Submitted. [PDF] 22.    The Koopman representation and positive Rokhlin entropy Submitted. [PDF] 21.    Continuous combinatorics of abelian group actions (with S. Gao, S. Jackson, and E. Krohne) Submitted. [PDF] 20.    Positive entropy actions of countable groups factor onto Bernoulli shifts To appear in Journal of the American Mathematical Society. [PDF] 19.    Krieger's finite generator theorem for actions of countable groups III (with A. Alpeev) Submitted. [PDF] 18.    Folner tilings for actions of amenable groups (with C. Conley, S. Jackson, D. Kerr, A. Marks, and R. Tucker-Drob) Mathematische Annalen 371 (2018), no. 1-2, 663-683. [PDF] 17.    Hyperfiniteness and Borel combinatorics To appear in Journal of the European Mathematical Society. [PDF] 16.    Weak containment and Rokhlin entropy Submitted. [PDF] 15.    Cost, l^2-Betti numbers, and the sofic entropy of some algebraic actions (with Damien Gaboriau) To appear in Journal d'Analyse Mathematique. [PDF] 14.    Forcing constructions and countable Borel equivalence relations (with Su Gao, Steve Jackson, and Edward Krohne) Submitted. [PDF] 13.    Krieger's finite generator theorem for actions of countable groups II To appear in Journal of Modern Dynamics. [PDF] 12.    Krieger's finite generator theorem for actions of countable groups I To appear in Inventiones Mathematicae. [PDF] 11.    Borel structurability on the 2-shift of a countable group (with Robin D. Tucker-Drob) Annals of Pure and Applied Logic 167 (2016), no. 1, 1-21. [PDF] 10.    Every action of a non-amenable group is the factor of a small action Journal of Modern Dynamics 8 (2014), no. 2, 251-270. [PDF] 9.      Locally nilpotent groups and hyperfinite equivalence relations (with Scott Schneider) Submitted. [PDF] 8.      Arbitrarily large residual finiteness growth (with Khalid Bou-Rabee) To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal). [PDF] 7.      Ergodic actions of countable groups and finite generating partitions Groups, Geometry, and Dynamics 9 (2015), no. 3, 793-810. [PDF] 6.      Finite entropy actions of free groups, rigidity of stabilizers, and a Howe--Moore type phenomenon Journal d'Analyse Mathematique 129 (2016), no. 1, 309-340. [PDF] 5.      A subgroup formula for f-invariant entropy Ergodic Theory and Dynamical Systems 34 (2014), no. 1, 263-298. [PDF] 4.      Group colorings and Bernoulli subflows (with Su Gao and Steve Jackson) Memoirs of the American Mathematical Society 241 (2016), no. 1141, 1-241. [Preprint] 3.      Generalizing Magnus' characterization of free groups to some free products (with Khalid Bou-Rabee) Communications in Algebra 42 (2014), no. 9, 3950-3962. [PDF] 2.      Burnside's problem, spanning trees, and tilings Geometry & Topology 18 (2014), no. 1, 179-210. [PDF] 1.      A coloring property for countable groups (with Su Gao and Steve Jackson) Mathematical Proceedings of the Cambridge Philosophical Society 147 (2009), no. 3, 579-592. [PDF] Abstracts 23.    Bernoulli shifts with bases of equal entropy are isomorphic Submitted. [PDF] We prove that if G is a countably infinite group and (L, \lambda) and (K, \kappa) are probability spaces having equal Shannon entropy, then the Bernoulli shifts (L^G, \lambda^G) and (K^G, \kappa^G) are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both \lambda and \kappa have at least 3 points in their support. We furthermore produce finitary isomorphisms in the case where both L and K are finite. 22.    The Koopman representation and positive Rokhlin entropy Submitted. [PDF] In a prior paper, the author generalized the classical factor theorem of Sinai to actions of arbitrary countably infinite groups. In the present paper, we use this theorem and the techniques of its proof in order to study connections between positive entropy phenomena and the Koopman representation. Following the line of work initiated by Hayes for sofic entropy, we show in a certain precise manner that all positive entropy must come from portions of the Koopman representation that embed into the left-regular representation. By combining this result with the generalized factor theorem of the previous paper, we conclude that for actions having completely positive outer entropy, the Koopman representation must be isomorphic to the countable direct sum of the left-regular representation. This generalizes a theorem of Dooley--Golodets for countable amenable groups. As a final consequence, we observe that actions with completely positive outer entropy must be mixing, and when the group is non-amenable they must be strongly ergodic and have spectral gap. We also prove generalizations of these results that apply relative to sub-sigma-algebras and apply to non-free actions. 21.    Continuous combinatorics of abelian group actions (with S. Gao, S. Jackson, and E. Krohne) Submitted. [PDF] This paper develops techniques which are used to answer a number of questions in the theory of equivalence relations generated by continuous actions of abelian groups. The methods center around the construction of certain specialized hyper-aperiodic elements, which produce compact subflows with useful properties. For example, we show that there is no continuous 3-coloring of the Cayley graph on $F(2^{\Z^2})$, the free part of the shift action of $\Z^2$ on $2^{\Z^2}$. With earlier work of the authors this computes the continuous chromatic number of $F(2^{\Z^2})$ to be exactly 4. Combined with marker arguments for the positive directions, our methods allow us to analyze continuous homomorphisms into graphs, and more generally equivariant maps into subshifts of finite type. We present a general construction of a finite set of "tiles'' for  $2^{\Z^n}$ (there are 12 for n=2) such that questions about the existence of continuous homomorphisms into various structures reduce to finitary combinatorial questions about the tiles. This tile analysis is used to deduce a number of results about $F(2^{\Z^n})$. 20.    Positive entropy actions of countable groups factor onto Bernoulli shifts To appear in Journal of the American Mathematical Society. [PDF] We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy) then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory. 19.    Krieger's finite generator theorem for actions of countable groups III (with A. Alpeev) Submitted. [PDF] We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov--Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers. 18.    Folner tilings for actions of amenable groups (with C. Conley, S. Jackson, D. Kerr, A. Marks, and R. Tucker-Drob) Mathematische Annalen 371 (2018), no. 1-2, 663-683. [PDF] We show that every probability-measure-preserving action of a countable amenable group G can be tiled, modulo a null set, using finitely many finite subsets of G ("shapes'') with prescribed approximate invariance so that the collection of tiling centers for each shape is Borel. This is a dynamical version of the Downarowicz--Huczek--Zhang tiling theorem for countable amenable groups and strengthens the Ornstein--Weiss Rokhlin lemma. As an application we prove that, for every countably infinite amenable group G, the crossed product of a generic free minimal action of G on the Cantor set is Z-stable. 17.    Hyperfiniteness and Borel combinatorics To appear in Journal of the European Mathematical Society. [PDF] We study the relationship between hyperfiniteness and problems in Borel graph combinatorics by adapting game-theoretic techniques introduced by Marks to the hyperfinite setting. We compute the possible Borel chromatic numbers and edge chromatic numbers of bounded degree acyclic hyperfinite Borel graphs and use this to answer a question of Kechris and Marks about the relationship between Borel chromatic number and measure chromatic number. We also show that for every d > 1 there is a d-regular acyclic hyperfinite Borel bipartite graph with no Borel perfect matching. These techniques also give examples of hyperfinite bounded degree Borel graphs for which the Borel local lemma fails, in contrast to the recent results of Csoka, Grabowski, Mathe, Pikhurko, and Tyros. Related to the Borel Ruziewicz problem, we show there is a continuous paradoxical action of $(\Z/2\Z)^{*3}$ on a Polish space that admits a finitely additive invariant Borel probability measure, but admits no countably additive invariant Borel probability measure. In the context of studying ultrafilters on the quotient space of equivalence relations under AD, we also construct an ultrafilter U on the quotient of $E_0$ which has surprising complexity. In particular, Martin's measure is Rudin-Kiesler reducible to U. We end with a problem about whether every hyperfinite bounded degree Borel graph has a witness to its hyperfiniteness which is uniformly bounded below in size. 16.    Weak containment and Rokhlin entropy Submitted. [PDF] We define a new notion of weak containment for joinings, and we show that this notion implies an inequality between relative Rokhlin entropies. This leads to new upper bounds to Rokhlin entropy. We also use this notion to study how Pinsker algebras behave under direct products, and we study the Rokhlin entropy of restricted actions of finite-index subgroups. 15.    Cost, l^2-Betti numbers, and the sofic entropy of some algebraic actions (with Damien Gaboriau) To appear in Journal d'Analyse Mathematique. [PDF] In 1987, Ornstein and Weiss discovered that the Bernoulli 2-shift over the rank two free group factors onto the seemingly larger Bernoulli 4-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first l^2-Betti number of the group. More generally, we study coboundary maps arising from simplicial actions and, under certain assumptions, relate l^2-Betti numbers to the failure of the Juzvinskii addition formula. This work is built upon a study of entropy theory for algebraic actions. We prove that for actions on profinite groups via continuous group automorphisms, topological sofic entropy is equal to measure sofic entropy with respect to Haar measure whenever the homoclinic subgroup is dense. For algebraic actions of residually finite groups we find sufficient conditions for the sofic entropy to be equal to the supremum exponential growth rate of periodic points. 14.    Forcing constructions and countable Borel equivalence relations (with Su Gao, Steve Jackson, and Edward Krohne) Submitted. [PDF]We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.     13.    Krieger's finite generator theorem for actions of countable groups II To appear in Journal of Modern Dynamics. [PDF]We continue the study of Rokhlin entropy, an isomorphism invariant for ergodic p.m.p. actions of general countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abert--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin entropy of a Bernoulli shift is equal to the Shannon entropy of its base; (ii) Bernoulli shifts have completely positive Rokhlin entropy; and (iii) Gottschalk's surjunctivity conjecture and Kaplansky's direct finiteness conjecture are true.        12.    Krieger's finite generator theorem for actions of countable groups I To appear in Inventiones Mathematicae. [PDF]For an ergodic p.m.p. action G \acts (X, \mu) of a countable group G, we define the Rokhlin entropy h_G^{Rok}(X, \mu) to be the infimum of the Shannon entropies of countable generating partitions. It is known that for free ergodic actions of amenable groups this notion coincides with classical Kolmogorov--Sinai entropy. It is thus natural to view Rokhlin entropy as a close analogue to classical entropy. Under this analogy we prove that Krieger's finite generator theorem holds for all countably infinite groups. Specifically, if h_G^{Rok}(X, \mu) < log(k) then there exists a generating partition consisting of k sets. We actually obtain a notably stronger result which is new even in the case of actions by the integers. Our proofs are entirely self-contained and do not rely on the original Krieger theorem.        11.    Borel structurability on the 2-shift of a countable group (with Robin D. Tucker-Drob) Annals of Pure and Applied Logic 167 (2016), no. 1, 1-21. [PDF]We show that for any infinite countable group G and for any free Borel action of G on X there exists an equivariant class-bijective Borel map from X to the free part Free(2^G) of the 2-shift 2^G. This implies that any Borel structurability which holds for the equivalence relation generated by G acting on Free(2^G) must hold a fortiori for all equivalence relations coming from free Borel actions of G. A related consequence is that the Borel chromatic number of Free(2^G) is the maximum among Borel chromatic numbers of free actions of G. This answers a question of Marks. Our construction is flexible and, using an appropriate notion of genericity, we are able to show that in fact the generic G-equivariant map to 2^G lands in the free part. As a corollary we obtain that for every \epsilon > 0, every free p.m.p. action of G has a free factor which admits a 2-piece generating partition with Shannon entropy less than \epsilon. This generalizes a result of Danilenko and Park.    10.    Every action of a non-amenable group is the factor of a small action Journal of Modern Dynamics 8 (2014), no. 2, 251-270. [PDF]It is well known that if G is a countable amenable group and G \acts (Y, \nu) factors onto G \acts (X, \mu), then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if G \acts (X, \mu) has infinite entropy, then the action G \acts (Y, \nu) does not admit any finite generating partition. On the other hand, we prove that if G is a countable non-amenable group then there exists a finite integer n with the following property: for every probability-measure-preserving action G \acts (X, \mu) there is a G-invariant probability measure \nu on n^G such that G \acts (n^G, \nu) factors onto G \acts (X, \mu). For many non-amenable groups, n can be chosen to be 4 or smaller. We also obtain a similar result with respect to continuous actions on compact spaces and continuous factor maps.  9.      Locally nilpotent groups and hyperfinite equivalence relations (with Scott Schneider) Submitted. [PDF]A long standing open problem in the theory of hyperfinite equivalence relations asks if the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper we show that this question has a positive answer when the acting group is locally nilpotent. This extends previous results obtained by Gao--Jackson for abelian groups and by Jackson--Kechris--Louveau for finitely generated nilpotent-by-finite groups. Our proof is based on a mixture of coarse geometric properties of locally nilpotent groups together with an adaptation of the Gao--Jackson machinery.  8.      Arbitrarily large residual finiteness growth (with Khalid Bou-Rabee) To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal). [PDF]The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship between residual finiteness growth and some decision problems in groups, which we apply to our new groups.  7.      Ergodic actions of countable groups and finite generating partitions Groups, Geometry, and Dynamics 9 (2015), no. 3, 793-810. [PDF]We prove the following finite generator theorem. Let G be a countable group acting ergodically on a standard probability space. Suppose this action admits a generating partition having finite Shannon entropy. Then the action admits a finite generating partition. We also discuss relationships between generating partitions and f-invariant and sofic entropies.  6.      Finite entropy actions of free groups, rigidity of stabilizers, and a Howe--Moore type phenomenon Journal d'Analyse Mathematique 129 (2016), no. 1, 309-340. [PDF]We study a notion of entropy for probability measure preserving actions of finitely generated free groups, called f-invariant entropy, introduced by Lewis Bowen. In the degenerate case, the f-invariant entropy is negative infinity. In this paper we investigate the qualitative consequences of having finite f-invariant entropy. We find three main properties of such actions. First, the stabilizers occurring in factors of such actions are highly restricted. Specifically, the stabilizer of almost every point must be either trivial or of finite index. Second, such actions are very chaotic in the sense that, when the space is not essentially countable, every non-identity group element acts with infinite Kolmogorov--Sinai entropy. Finally, we show that such actions display behavior reminiscent of the Howe--Moore property. Specifically, if the action is ergodic then there is an integer n such that for every non-trivial normal subgroup K the number of K-ergodic components is at most n. Our results are based on a new formula for f-invariant entropy.  5.      A subgroup formula for f-invariant entropy Ergodic Theory and Dynamical Systems 34 (2014), no. 1, 263-298. [PDF]We study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by Lewis Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well known formula for the Kolmogorov--Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.  4.      Group colorings and Bernoulli subflows (with Su Gao and Steve Jackson) Memoirs of the American Mathematical Society 241 (2016), no. 1141, 1-241. [Preprint]In this paper we study the dynamics of Bernoulli flows and their subflows over general countable groups from the symbolic and topological perspectives. We study free subflows (subflows in which every point has trivial stabilizer), minimal subflows, disjointness of subflows, the problem of classifying subflows up to topological conjugacy, and the differences in dynamical behavior between pairs of points which disagree on finitely many coordinates. We call a point hyper aperiodic if the closure of its orbit is a free subflow and we call it minimal if the closure of its orbit is a minimal subflow.We prove that the set of all (minimal) hyper aperiodic points is always dense but also meager and null. By employing notions and ideas from descriptive set theory, we study the complexity of the sets of hyper aperiodic points and of minimal points and completely determine their descriptive complexity. In doing this we introduce a new notion of countable flecc groups and study their properties. We obtain a dichotomy for the complexity of classifying free subflows up to topological conjugacy. For locally finite groups the topological conjugacy relation for all (free) subflows is hyperfinite and nonsmooth. For nonlocally finite groups the relation is Borel bireducible with the universal countable Borel equivalence relation.A primary focus of the paper is to develop constructive methods for the notions studied. To construct hyper aperiodic points, a fundamental method of construction of multi-layer marker structures is developed with great generality. Variations of the fundamental method are used in many proofs in the paper, and we expect them to be useful more broadly in geometric group theory. As a special case of such marker structures, we study the notion of ccc groups and prove the ccc-ness for countable nilpotent, polycyclic, residually finite, locally finite groups and for free products.  3.      Generalizing Magnus' characterization of free groups to some free products (with Khalid Bou-Rabee) Communications in Algebra 42 (2014), no. 9, 3950-3962. [PDF]A residually nilpotent group is k-parafree if all of its lower central series quotients match those of a free group of rank k. Magnus proved that k-parafree groups of rank k are themselves free. In this note we mimic this theory with finite extensions of free groups, with an emphasis on free products of the cyclic group C_p, for p an odd prime. We show that for n \leq p Magnus' characterization holds for the n-fold free product C_p^{*n} within the class of finite-extensions of free groups. Specifically, if n \leq p and G is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as C_p^{*n}, then G \cong C_p^{*n}. We also show that such a characterization does not hold in the class of finitely generated groups. That is, we construct a rank 2 residually nilpotent group G that shares all its lower central series quotients with C_p * C_p, but is not C_p * C_p.  2.      Burnside's problem, spanning trees, and tilings Geometry & Topology 18 (2014), no. 1, 179-210. [PDF]In this paper we study geometric versions of Burnside's Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is non-amenable if and only if it admits a translation-like action by any (equivalently every) non-abelian free group. We strengthen Whyte's result by proving that this translation-like action can be chosen to be transitive when the acting free group is finitely generated. We furthermore prove that the geometric version of Burnside's Problem holds true. That is, every finitely generated infinite group admits a translation-like action by Z. This answers a question posed by Whyte. In pursuit of these results we discover an interesting property of Cayley graphs: every finitely generated infinite group G has some Cayley graph having a regular spanning tree. This regular spanning tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian path) if and only if G has finitely many ends, and it can be chosen to have any degree greater than 2 if and only if G is non-amenable. We use this last result to then study tilings of groups. We define a general notion of polytilings and extend the notion of MT groups and ccc groups to the setting of polytilings. We prove that every countable group is poly-MT and every finitely generated group is poly-ccc.  1.      A coloring property for countable groups (with Su Gao and Steve Jackson) Mathematical Proceedings of the Cambridge Philosophical Society 147 (2009), no. 3, 579-592. [PDF]Motivated by research on hyperfinite equivalence relations we define a coloring property for countable groups. We prove that every countable group has the coloring property. This implies a compactness theorem for closed complete sections of the free part of the shift action of G on 2^G. Our theorems generalize known results about Z.  Webpage updated Oct. 2018