Brandon Seward


PhD Candidate
Department of Mathematics
University of Michigan
2848 East Hall
530 Church Street
Ann Arbor, MI 48109
USA

bseward@umich.edu

Curriculum Vitae





I am a fourth year PhD student at the University of Michigan studying pure mathematics. I was previously at the University of North Texas where I recieved Bachelor's and Master's degrees in mathematics. I am currently being supported by an NSF graduate research fellowship. My mathematical interests include geometric and combinatorial group theory, group actions, ergodic theory, symbolic dynamics, topological dynamics, Bernoulli shifts, descriptive set theory, and the theory of countable Borel equivalence relations. My advisor is Ralf Spatzier.

Papers:
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.
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.
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. Our results are based on a new formula for f-invariant entropy. Our main theorem is that 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. We also find that such actions must have only countably many ergodic components, and when the space is not essentially countable every non-trivial group element must act with infinite Kolmogorov--Sinai entropy. We also 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.
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.
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.
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 $C_p * C_p$.
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.
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.

Other:
  • "On the density of minimal free subflows of general symbolic flows," Master's Thesis, University of North Texas, August 2009. [PDF]
This thesis was never published but instead the content was incorporated into the paper "Group colorings and Bernoulli subflows" (joint with Su Gao and Steve Jackson) found above.
  • List of Preliminary Exam Topics. April 29, 2011. [PDF]
The Preliminary Exam is an oral exam on the area of math in which the student intends to specialize. It is the final requirement for obtaining candidate status.
Updated May 2013