Brandon Sewardbseward@umich.eduPhD CandidateDepartment of MathematicsUniversity of Michigan2848 East Hall530 Church StreetAnn Arbor, MI 48109USACurriculum VitaeResearch StatementTeaching Statement My research falls into the areas of group theory, dynamical systems, and descriptive set theory.Much of my work is in ergodic theory, in particular entropy theory. Frequently, there are strong connections between groups and the properties of their actions. I enjoy many topics in group theory and enjoy studying countable groups themselves. Likewise, there are many connections between ergodic theory, topological dynamics, and Borel dynamics, and I have interest in all three.My advisor is Ralf Spatzier.My wife is an MFA graduate student in dance here at the University of Michigan. Here is her webpage. Publications and Preprints12.    Krieger's finite generator theorem for ergodic actions of countable groups IPreprint. [PDF]11.    Borel structurability on the 2-shift of a countable group (with Robin D. Tucker-Drob)Submitted. [PDF]10.    Every action of a non-amenable group is the factor of a small actionTo appear in Journal of Modern Dynamics. [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 partitionsTo appear in Groups, Geometry, and Dynamics. [PDF]6.      Finite entropy actions of free groups, rigidity of stabilizers, and a Howe--Moore type phenomenonTo appear in Journal d'Analyse Mathematique. [PDF]5.      A subgroup formula for f-invariant entropyErgodic Theory and Dynamical Systems 34 (2014), no. 1, 263-298. [PDF]4.      Group colorings and Bernoulli subflows (with Su Gao and Steve Jackson)To appear in Memoirs of the American Mathematical Society. [PDF]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 tilingsGeometry & 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]Abstracts12.    Krieger's finite generator theorem for ergodic actions of countable groups IPreprint. [PDF]For an ergodic p.m.p. action G \acts (X, \mu) of a countable group G, we consider the infimum h_G^{Rok}(X, \mu) of the Shannon entropies of countable generating partitions. In the case of free ergodic actions of amenable groups it is known that this quantity is equal to the entropy of the action. It is thus natural to view h_G^{Rok}(X, \mu) as a close analogue to 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. Furthermore, we develop formulas for h_G^{Rok}(X, \mu) which are analogues to the classical Kolmogorov and Kolmogorov--Sinai theorems from entropy theory. 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)Sumbitted. [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 actionTo appear in Journal of Modern Dynamics. [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 partitionsTo appear in Groups, Geometry, and Dynamics. [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 phenomenonTo appear in Journal d'Analyse Mathematique. [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 entropyErgodic 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)To appear in Memoirs of the American Mathematical Society. [PDF]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 tilingsGeometry & 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 Aug. 2014