scholarly journals Revisiting Leighton’s theorem with the Haar measure

2020 ◽  
pp. 1-9
Author(s):  
DANIEL J. WOODHOUSE
Keyword(s):  
Haar Measure ◽  
Free Groups ◽  
Finite Cover ◽  
Covering Theorem ◽  
Rigidity Result ◽  
Graph Covering ◽  
Finite Graphs ◽  
Universal Covers ◽  
Line Patterns

Abstract Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

Random Colourings and Automorphism Breaking in Locally Finite Graphs

2013 ◽  
Vol 22 (6) ◽  
pp. 885-909 ◽  
Author(s):  
FLORIAN LEHNER
Keyword(s):  
Automorphism Group ◽  
Haar Measure ◽  
Finite Graph ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graph ◽  
Locally Finite Graphs ◽  
Null Set

A colouring of a graphGis called distinguishing if its stabilizer in AutGis trivial. It has been conjectured that, if every automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We study properties of random 2-colourings of locally finite graphs and show that the stabilizer of such a colouring is almost surely nowhere dense in AutGand a null set with respect to the Haar measure on the automorphism group. We also investigate random 2-colourings in several classes of locally finite graphs where the existence of a distinguishing 2-colouring has already been established. It turns out that in all of these cases a random 2-colouring is almost surely distinguishing.

scholarly journals Splitting line patterns in free groups

2016 ◽  
Vol 16 (2) ◽  
pp. 621-673 ◽  
Author(s):  
Christopher H Cashen
Keyword(s):  
Free Groups ◽  
Line Patterns

scholarly journals Diffeomorphism group valued cocycles over higher-rank abelian Anosov actions

2018 ◽  
Vol 40 (1) ◽  
pp. 117-141 ◽  
Author(s):  
DANIJELA DAMJANOVIĆ ◽  
DISHENG XU
Keyword(s):  
Fixed Point ◽  
Universal Cover ◽  
Diffeomorphism Group ◽  
Finite Cover ◽  
Diffeomorphism Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Rigidity Results ◽  
Anosov Actions ◽  
Higher Rank

We prove that every smooth diffeomorphism group valued cocycle over certain$\mathbb{Z}^{k}$Anosov actions on tori (and more generally on infranilmanifolds) is a smooth coboundary on a finite cover, if the cocycle is center bunched and trivial at a fixed point. For smooth cocycles which are not trivial at a fixed point, we have smooth reduction of cocycles to constant ones, when lifted to the universal cover. These results on cocycle trivialization apply, via the existing global rigidity results, to maximal Cartan$\mathbb{Z}^{k}$($k\geq 3$) actions by Anosov diffeomorphisms (with at least one transitive), on any compact smooth manifold. This is the first rigidity result for cocycles over$\mathbb{Z}^{k}$actions with values in diffeomorphism groups which does not require any restrictions on the smallness of the cocycle or on the diffeomorphism group.

scholarly journals Folding-like techniques for CAT(0) cube complexes

2021 ◽  
pp. 1-12
Author(s):  
MICHAEL BEN–ZVI ◽  
ROBERT KROPHOLLER ◽  
RYLEE ALANZA LYMAN
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Coxeter Groups ◽  
Free Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Seminal Paper ◽  
Finite Graphs ◽  
Cube Complexes ◽  
Positively Curved

Abstract In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

scholarly journals Line patterns in free groups

2011 ◽  
Vol 15 (3) ◽  
pp. 1419-1475 ◽  
Author(s):  
Christopher H Cashen ◽  
Nataša Macura
Keyword(s):  
Free Groups ◽  
Line Patterns

scholarly journals On Anosov diffeomorphisms with asymptotically conformal periodic data

2009 ◽  
Vol 29 (1) ◽  
pp. 117-136 ◽  
Author(s):  
BORIS KALININ ◽  
VICTORIA SADOVSKAYA
Keyword(s):  
Periodic Points ◽  
Finite Cover ◽  
Rigidity Result ◽  
Anosov Diffeomorphisms ◽  
Local Rigidity ◽  
Return Maps ◽  
Periodic Data ◽  
Conformal Structures ◽  
Derivatives Of ◽  
Volume Preserving

AbstractWe consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We prove various properties of such systems, including strong pinching, C1+β smoothness of the Anosov splitting, and C1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume-preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of 𝕋4; as a corollary, we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

The usefulness of high index low symmetry zone axes for holz line analysis

1987 ◽  
Vol 45 ◽  
pp. 384-385
Author(s):  
C. M. Sung ◽  
D. B. Williams
Keyword(s):  
Lattice Parameter ◽  
High Order ◽  
Zone Axis ◽  
High Index ◽  
High Symmetry ◽  
Second Phases ◽  
Line Patterns ◽  
Low Symmetry ◽  
Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

Imaging nanostructures on ablated kapton polymide

1993 ◽  
Vol 51 ◽  
pp. 872-873
Author(s):  
Daniel L. Callahan ◽  
H. M. Phillips ◽  
R. Sauerbrey
Keyword(s):  
Electron Microscopy ◽  
Polymer Surface ◽  
Grating Period ◽  
Laser Wavelength ◽  
Integrated Devices ◽  
Transmission Electron ◽  
Excimer Laser Irradiation ◽  
Produce Line ◽  
Line Patterns ◽  
Anisotropic Conduction

Excimer laser irradiation has been used to interferometrically ablate submicron line patterns on to Kapton polyimide. Such patterned material may exhibit highly anisotropic conduction as was predicted from previous studies showing enhanced conductivity from uniformly ablated material. We are currently exploiting this phenomenon to create integrated devices using conventional polymers as both dielectrics and conductors. Extensive scanning electron microscopy (SEM) and limited transmission electron microscopy (TEM) have been conducted in order to characterize the morphology of such patterned nanostructures as a function of processing conditions.The ablation technique employed produces an interference pattern on the polymer surface of period equal to half that of a diffraction grating period, independent of the laser wavelength. In these experiments, a 328 nm grating has been used to produce line patterns of 164 nm line-spacings as shown in Figures 1 and 2. A 200 Å Au coating has been used to both prevent charging and, perhaps more importantly, enhance contrast.

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