scholarly journals Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles

2017 ◽  
Vol 2019 (13) ◽  
pp. 4004-4046
Author(s):  
Corey Bregman
Keyword(s):  
Unit Circle ◽  
Dimensional Torus ◽  
Torus Bundle ◽  
Generating Set ◽  
Growth Series ◽  
Torus Bundles ◽  
Almost Convex ◽  
Higher Dimensional ◽  
Finite Index Subgroup ◽  
Almost Convexity

AbstractGiven a matrix $A\in SL(N,\mathbb{Z})$, form the semidirect product $G=\mathbb{Z}^N\rtimes_A \mathbb{Z}$ where the $\mathbb{Z}$-factor acts on $\mathbb{Z}^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this article, we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $H\leq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.

scholarly journals ON THE TOPOLOGY OF T-DUALITY

2005 ◽  
Vol 17 (01) ◽  
pp. 77-112 ◽  
Author(s):  
ULRICH BUNKE ◽  
THOMAS SCHICK
Keyword(s):  
Homotopy Type ◽  
Cohomology Class ◽  
Dimensional Torus ◽  
Duality Transformation ◽  
Classifying Space ◽  
Simple Derivation ◽  
Torus Bundles ◽  
Twisted Cohomology ◽  
Higher Dimensional ◽  
Topological Version

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

A Cayley graph for F2 × F2 which is not minimally almost convex

2021 ◽  
pp. 1-12
Author(s):  
Andrew Elvey Price
Keyword(s):  
Cayley Graph ◽  
The Other ◽  
Fellow Traveler ◽  
Generating Set ◽  
Other Hand ◽  
Almost Convex ◽  
Graph Property ◽  
Almost Convexity

We give an example of a Cayley graph [Formula: see text] for the group [Formula: see text] which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for [Formula: see text] does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property [Formula: see text] lying between FFTP and MAC (i.e., [Formula: see text]) is dependent on the generating set. This includes the well-known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru’s condition [Formula: see text] and the basepoint loop shortening property (LSP) for which dependence on the generating set was previously unknown. We also show that the Cayley graph [Formula: see text] does not have the LSP, so this property also depends on the generating set.

scholarly journals Homological mirror symmetry for higher-dimensional pairs of pants

2020 ◽  
Vol 156 (7) ◽  
pp. 1310-1347
Author(s):  
Yankı Lekili ◽  
Alexander Polishchuk
Keyword(s):  
Mirror Symmetry ◽  
Derived Category ◽  
Affine Variety ◽  
Fukaya Category ◽  
Homological Mirror Symmetry ◽  
Endomorphism Algebra ◽  
Generating Set ◽  
Fukaya Categories ◽  
Higher Dimensional ◽  
Abelian Covers

Using Auroux’s description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of $k+1$ generic hyperplanes in $\mathbb{CP}^{n}$, for $k\geqslant n$, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of $n+2$ generic hyperplanes in $\mathbb{C}P^{n}$ ($n$-dimensional pair of pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety $x_{1}x_{2}\ldots x_{n+1}=0$. By localizing, we deduce that the (fully) wrapped Fukaya category of the $n$-dimensional pair of pants is equivalent to the derived category of $x_{1}x_{2}\ldots x_{n+1}=0$. We also prove similar equivalences for finite abelian covers of the $n$-dimensional pair of pants.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

scholarly journals A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

2015 ◽  
Vol 69 (1) ◽  
pp. 109
Author(s):  
Grzegorz Zborowski
Keyword(s):  
Ricci Tensor ◽  
Torus Bundle ◽  
Torus Bundles

An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies ∇<sub>X</sub> Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds.

Growth of Groups

2017 ◽  
Author(s):  
Eric Freden
Keyword(s):  
Word Length ◽  
Polynomial Growth ◽  
Formal Languages ◽  
Research Projects ◽  
Growth Theorem ◽  
Generating Set ◽  
Growth Series ◽  
Growth Of Groups ◽  
Counting Methods ◽  
Context Free

This chapter considers the growth of a group. It begins with the case of a group with a given finite generating set, in which the ball B(1, r) of radius r centered at the identity 1 is the set of group elements whose word length is less than or equal to r. Given a group that is fixed and some r greater than or equal to 0, the chapter invokes Gromov's polynomial growth theorem to determine how many group elements are in B(1, r) and how many group elements are in S(1, r). It also explores the use of simple counting methods to compute several growth series before concluding with an overview of cone types, formal languages and context-free grammars, and the DSV method used to compute the growth of grammar productions. The discussion includes exercises and research projects.

scholarly journals COMPUTING WORD LENGTH IN ALTERNATE PRESENTATIONS OF THOMPSON'S GROUP F

2009 ◽  
Vol 19 (08) ◽  
pp. 963-997 ◽  
Author(s):  
MATTHEW HORAK ◽  
MELANIE STEIN ◽  
JENNIFER TABACK
Keyword(s):  
Word Length ◽  
New Method ◽  
Generating Set ◽  
Almost Convex ◽  
Thompson’S Group ◽  
Thompson's Group F

We introduce a new method for computing the word length of an element of Thompson's group F with respect to a "consecutive" generating set of the form Xn = {x0,x1, …,xn}, which is a subset of the standard infinite generating set for F. We use this method to show that (F, Xn) is not almost convex, and has pockets of increasing, though bounded, depth dependent on n.

scholarly journals Homological Percolation: The Formation of Giant k-Cycles

2020 ◽  
Author(s):  
Omer Bobrowski ◽  
Primoz Skraba
Keyword(s):  
Thermodynamic Limit ◽  
Exponential Decay ◽  
Algebraic Topology ◽  
Percolation Model ◽  
Critical Values ◽  
Giant Component ◽  
Connected Components ◽  
Continuum Percolation ◽  
Dimensional Torus ◽  
Higher Dimensional

Abstract In this paper we introduce and study a higher dimensional analogue of the giant component in continuum percolation. Using the language of algebraic topology, we define the notion of giant $k$-dimensional cycles (with $0$-cycles being connected components). Considering a continuum percolation model in the flat $d$-dimensional torus, we show that all the giant $k$-cycles ($1\le k \le d-1$) appear in the regime known as the thermodynamic limit. We also prove that the thresholds for the emergence of the giant $k$-cycles are increasing in $k$ and are tightly related to the critical values in continuum percolation. Finally, we provide bounds for the exponential decay of the probabilities of giant cycles appearing.

The Growth Series of Compact Hyperbolic Coxeter Groups with 4 and 5 Generators

1998 ◽  
Vol 41 (2) ◽  
pp. 231-239 ◽  
Author(s):  
R. L. Worthington
Keyword(s):  
Unit Circle ◽  
Coxeter Groups ◽  
Growth Series

AbstractThe growth series of compact hyperbolic Coxeter groups with 4 and 5 generators are explicitly calculated. The assertions of J. Cannon and Ph. Wagreich for the 4-generated groups, that the poles of the growth series lie on the unit circle, with the exception of a single real reciprocal pair of poles, are verified. We also verify that for the 5-generated groups, this phenomenon fails.

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