Homological eigenvalues of lifts of pseudo-Anosov mapping classes to finite covers

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

Download Full-text

Permutation groups with small orbit growth

Journal of Group Theory ◽

10.1515/jgth-2018-0220 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Manuel Bodirsky ◽

Bertalan Bodor

Keyword(s):

Automorphism Group ◽

Constraint Satisfaction ◽

Constraint Satisfaction Problem ◽

Permutation Groups ◽

First Order ◽

Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

Download Full-text

Constructing 1-cusped isospectral non-isometric hyperbolic 3-manifolds

Journal of Topology and Analysis ◽

10.1142/s1793525318500024 ◽

2017 ◽

Vol 10 (01) ◽

pp. 1-25

Author(s):

Stavros Garoufalidis ◽

Alan W. Reid

Keyword(s):

Discrete Spectrum ◽

Eisenstein Series ◽

Finite Volume ◽

Finite Covers ◽

Figure Eight Knot ◽

Complex Length

We construct infinitely many examples of pairs of isospectral but non-isometric [Formula: see text]-cusped hyperbolic [Formula: see text]-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an application of Sunada’s method in the cusped setting, and so in addition our pairs are finite covers of the same degree of a 1-cusped hyperbolic 3-orbifold (indeed manifold) and also have the same complex length spectra. Finally we prove that any finite volume hyperbolic 3-manifold isospectral to the figure-eight knot complement is homeomorphic to the figure-eight knot complement.

Download Full-text

Finite Covers of 3-Manifolds Containing Essential Tori

Transactions of the American Mathematical Society ◽

10.2307/2001129 ◽

1988 ◽

Vol 310 (1) ◽

pp. 381 ◽

Cited By ~ 1

Author(s):

John Luecke

Keyword(s):

Finite Covers

Download Full-text

Deformation of finite morphisms and smoothing of ropes

Compositio Mathematica ◽

10.1112/s0010437x07003326 ◽

2008 ◽

Vol 144 (3) ◽

pp. 673-688 ◽

Cited By ~ 5

Author(s):

Francisco Javier Gallego ◽

Miguel González ◽

Bangere P. Purnaprajna

Keyword(s):

General Theory ◽

Projective Space ◽

General Result ◽

The Other ◽

Independent Interest ◽

Higher Dimension ◽

Smooth Curves ◽

Other Hand ◽

Finite Covers ◽

The One

AbstractIn this paper we prove that most ropes of arbitrary multiplicity supported on smooth curves can be smoothed. By a rope being smoothable we mean that the rope is the flat limit of a family of smooth, irreducible curves. To construct a smoothing, we connect, on the one hand, deformations of a finite morphism to projective space and, on the other hand, morphisms from a rope to projective space. We also prove a general result of independent interest, namely that finite covers onto smooth irreducible curves embedded in projective space can be deformed to a family of 1:1 maps. We apply our general theory to prove the smoothing of ropes of multiplicity 3 on P1. Even though this paper focuses on ropes of dimension 1, our method yields a general approach to deal with the smoothing of ropes of higher dimension.

Download Full-text