Searcharxiv⌕ Search

arXiv subjects

Aaron Abrams

Publications and source records attributed to Aaron Abrams.

30 records · Page 2Linked to original sources
arXiv2017

Fixed-energy harmonic functions

We study the map from conductances to edge energies for harmonic functions on finite graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. We call the associated function enharmonic. For rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. A consequence is the non-tileability of certain polygons by rational-area rectangles. For planar graphs there is an enharmonic conjugate function, together these form the real and imaginary parts of a "fixed energy" analytic function. In the planar scaling limit for ${\mathbb Z}^2$ (and the fixed south/west orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann equations, namely \begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*} We give an analog of the Riemann mapping theorem for these functions, as well as a variational approach to finding solutions in both the discrete and continuous settings.

math.PR↗
arXiv2016

Dull cut off for circulants

Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is measured in the total variation norm. This is a situation of bounded degree and no expansion. Sharp cut off or the cut off phenomenon has been shown to occur in families such as random walks on a hypercube [1] in which the degree is unbounded as well as on a random regular graph where the degree is fixed, but there is expansion [4]. Our examples agree with Peres' conjecture in [3] relating sharp cut off, spectral gap, and mixing time.

math.PR↗
arXiv2016

Sums of twisted circulants

The rate of convergence of simple random walk on the Heisenberg group over $Z/nZ$ with a standard generating set was determined by Bump et al [1,2]. We extend this result to random walks on the same groups with an arbitrary minimal symmetric generating set. We also determine the rate of convergence of simple random walk on higher-dimensional versions of the Heisenberg group with a standard generating set. We obtain our results via Fourier analysis, using an eigenvalue bound for sums of twisted circulant matrices. The key tool is a generalization of a version of the Heisenberg Uncertainty Principle due to Donoho-Stark [4].

math.PR↗
arXiv2016

Group trisections and smooth 4-manifolds

A trisection of a smooth, closed, oriented 4-manifold is a decomposition into three 4-dimensional 1-handlebodies meeting pairwise in 3-dimensional 1-handlebodies, with triple intersection a closed surface. The fundamental groups of the surface, the 3-dimensional handlebodies, the 4-dimensional handlebodies, and the closed 4-manifold, with homomorphisms between them induced by inclusion, form a commutative diagram of epimorphisms, which we call a trisection of the 4-manifold group. A trisected 4-manifold thus gives a trisected group; here we show that every trisected group uniquely determines a trisected 4-manifold. Together with Gay and Kirby's existence and uniqueness theorem for 4-manifold trisections, this gives a bijection from group trisections modulo isomorphism and a certain stabilization operation to smooth, closed, connected, oriented 4-manifolds modulo diffeomorphism. As a consequence, smooth 4-manifold topology is, in principle, entirely group theoretic. For example, the smooth 4-dimensional Poincaré conjecture can be reformulated as a purely group theoretic statement.

math.GT↗
arXiv2012

Homological and homotopical Dehn functions are different

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, while the homotopical Dehn function measures fillings of curves by disks. Since the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions, but prior to this work there were no known examples of finitely-presented groups for which the two functions differ. This paper gives the first such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina-Brady groups.

math.GR↗
arXiv2011

Discretized configurations and partial partitions

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.

math.GT↗
arXiv2011

Pushing fillings in right-angled Artin groups

We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties "at infinity," are defined here as a family of quasi-isometry invariants of groups; thus, these results give new information about the QI classification of RAAGs. By pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs called "orthoplex groups" which show that each of these upper bounds is sharp.

math.GR↗
arXiv2010

Distributions of order patterns of interval maps

A permutation $σ$ describing the relative orders of the first $n$ iterates of a point $x$ under a self-map $f$ of the interval $I=[0,1]$ is called an \emph{order pattern}. For fixed $f$ and $n$, measuring the points $x\in I$ (according to Lebesgue measure) that generate the order pattern $σ$ gives a probability distribution $μ_n(f)$ on the set of length $n$ permutations. We study the distributions that arise this way for various classes of functions $f$. Our main results treat the class of measure preserving functions. We obtain an exact description of the set of realizable distributions in this case: for each $n$ this set is a union of open faces of the polytope of flows on a certain digraph, and a simple combinatorial criterion determines which faces are included. We also show that for general $f$, apart from an obvious compatibility condition, there is no restriction on the sequence $\{μ_n(f)\}$ for $n=1,2,...$. In addition, we give a necessary condition for $f$ to have \emph{finite exclusion type}, i.e., for there to be finitely many order patterns that generate all order patterns not realized by $f$. Using entropy we show that if $f$ is piecewise continuous, piecewise monotone, and either ergodic or with points of arbitrarily high period, then $f$ cannot have finite exclusion type. This generalizes results of S. Elizalde.

math.CO↗
arXiv2004

Distances of Heegaard splittings

J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a fixed handlebody. With the same hypothesis we show the distance of the splitting (S,V, h^n(V)) grows linearly with n, answering a question of A Casson. In addition we prove the converse of Hempel's theorem. Our method is to study the action of h on the curve complex associated to S. We rely heavily on the result, due to H Masur and Y Minsky [Invent. Math. 1999], that the curve complex is Gromov hyperbolic.

math.GT↗
arXiv2004

Random Multiplication Approaches Uniform Measure in Finite Groups

In order to study how well a finite group might be generated by repeated random multiplications, P. Diaconis suggested the following urn model. An urn contains some balls labeled by elements which generate a group G. Two are drawn at random with replacement and a ball labeled with the group product (in the order they were picked) is added to the urn. We give a proof of his conjecture that the limiting fraction of balls labeled by each group element almost surely approaches 1/|G|.

math.PR↗
arXiv2001

Circles Minimize most Knot Energies

We define a new class of knot energies (known as renormalization energies) and prove that a broad class of these energies are uniquely minimized by the round circle. Most of O'Hara's knot energies belong to this class. This proves two conjectures of O'Hara and of Freedman, He, and Wang. We also find energies not minimized by a round circle. The proof is based on a theorem of G. Luko on average chord lengths of closed curves.

math.GT↗