1,721,033 research outputs found
Hyperbolic volume estimates via train tracks
Full text linkIn this thesis we describe how to estimate the distance spanned in the pants graph by a train track splitting sequence on a surface, up to multiplicative and additive constants. If some moderate assumptions on a splitting sequence are satisfied, each vertex set of a train track in it will represent a vertex of a graph which is naturally quasi-isometric to the pants graph; moreover the splitting sequence gives an edge-path in this graph so, more precisely, our distance estimate holds between the extreme points of this path. The present distance estimate is inspired by a result of Masur, Mosher and Schleimer for distances in the marking graph. However, we can apply their line of proof only after some manipulation of the splitting sequence: a rearrangement, changing the order the elementary moves are performed in, so that the ones producing Dehn twists are brought together; and then an untwisting, which suppresses the majority of these latter moves to give a new sequence, which does not end with the same track as before, but does not include any portion that is almost stationary in the pants graph. The required distance is then, up to constants, the number of splits occurring in the untwisted sequence. A consequence of our main theorem together with a result of Brock is that, given a pseudo-Anosov self-diffeomorphism Ï of a surface S, the maximal splitting sequence introduced by Agol gives us an estimate for the hyperbolic volume of the mapping torus built from S and Ï. There are also some interesting consequences for the hyperbolic volume of a solid torus minus a closed braid, via a machinery employed by Dynnikov and Wiest
Uniformly polynomial-time classification of surface homeomorphisms: Uniformly polynomial-time classification..
Full text linkWe describe an algorithm which, given two essential curves on a surface S, computes their distance in the curve graph of S, up to multiplicative and additive errors. As an application, we present an algorithm to decide the Nielsen–Thurston type (periodic, reducible, or pseudo-Anosov) of a mapping class of S. The novelty of our algorithms lies in the fact that their running time is polynomial in the size of the input and in the complexity of S—say, its Euler characteristic. This is in contrast with previously known algorithms, which run in polynomial time in the size of the input for any fixed surfaceS
Random irreducible quadrangulations
Full text linkIn the thesis we study random irreducible quadrangulations. We aim to extend known results on local limits of random planar maps and the perceived universality to the class of irreducible quadrangulations.
The first part of the thesis establishes a multitude of enumeration results using standard generating function techniques. We introduce various classes of irreducible quadrangulations and we make a detailed study of their generating functions and asymptotics. The material of this chapter forms the backbone of the subsequent calculations.
The last chapter is conceptually divided into two parts. The first part extends known sharp concentration results for maximum vertex degree and vertex degree distribution to the class of irreducible quadrangulations. In the second part we describe the skeleton decomposition of irreducible hulls and introduce a formal operator to evaluate statistical sums over the semi-layers of the decomposition.
The concluding section of the thesis exploits the properties of the hull operator to provide a new proof of the existence of the Uniform Infinite Irreducible Planar Quadrangulation (UIIPQ). We also explore the asymptotics of the iterates of the hull operator to derive limiting probabilities for the perimeter and volume of r-hulls in the UIIPQ. Our results are (unsurprisingly) in agreement with the conjectured universal behaviour observed in other random models such as the UIPT and UIPQ
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
The Kakimizu complex of a link
Full text linkWe study Seifert surfaces for links, and in particular the Kakimizu complex MS(L) of a link L, which is a simplicial complex that records the structure of the set of taut Seifert surfaces for L.First we study a connection between the reduced Alexander polynomial of a link and the uniqueness of taut Seifert surfaces. Specifically, we reprove and extend a particular case of a result of Juhasz, using very different methods, showing that if a non-split homogeneous link has a reduced Alexander polynomial whose constant term has modulus at most 3 then the link has a unique incompressible Seifert surface. More generally we see that this constant term controls the structure of any non-split homogeneous link.Next we give a complete proof of results stated by Hirasawa and Sakuma, describing explicitly the Kakimizu complex of any non-split, prime, special alternating link.We then calculate the form of the Kakimizu complex of a connected sum of two non-fibred links in terms of the Kakimizu complex of each of the two links. This has previously been done by Kakimizu when one of the two links is fibred.Finally, we address the question of when the Kakimizu complex is locally infinite. We show that if all the taut Seifert surfaces are connected then MS(L) can only be locally infinite when L is a satellite of a torus knot, a cable knot or a connected sum. Additionally we give examples of knots that exhibit this behaviour. We finish by showing that this picture is not complete when disconnected taut Seifert surfaces exist
Spectral properties of finite groups
Full text linkThis thesis concerns the diameter and spectral gap of finite groups. Our focus shall be on the asymptotic behaviour of these quantities for sequences of finite groups arising as quotients of a fixed infinite group. In Chapter 3 we give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups. In Chapter 4 we construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of SL2(𝔽p[t]) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in SL2(𝔽p[t]) Finally, in Chapter 5 we produce new expander congruence quotients of SL2 (ℤp), generalising work of Bourgain and Gamburd. The proof combines the Solovay-Kitaev procedure with a quantitative analysis of the algebraic geometry of these groups, which in turn relies on previously known examples of expanders
Profinite properties of 3-manifold groups
Full text linkIn this thesis we study the finite quotients of 3-manifold groups, concerning both residual properties of the groups and the properties of the 3-manifolds that can be detected using finite quotients of the fundamental group. A key theme is the analysis of when two 3-manifold groups can have the same families of finite quotients. We make a detailed study of this 'profinite rigidity' problem for Seifert fibre spaces and prove complete classification results for these manifolds. From Seifert fibre spaces we continue on this trajectory and extend our classification results to all graph manifolds. We illustrate this classification with examples and several consequences, including for graph knots and for mapping class groups. The third part of the thesis concerns the behaviour of the finite p-group quotients of 3-manifold groups. In general these quotients may be scarce and poorly behaved. We give results showing that some of these issues may be resolved by passing to finite-sheeted covers of the manifold involved. We also prove theorems concerning the p-conjugacy separability of certain graph manifold groups. The concluding chapter of the thesis collects other results linking low-dimensional topology and finite quotients of groups. In particular we prove that finite quotients of a right-angled Artin group distinguish it from other right-angled Artin groups, and we give an argument detecting the prime decomposition of certain 3-manifold groups from the finite p-group quotients
Four dimensional hyperbolic link complements via Kirby calculus
Full text linkThe primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S4 and a smooth standard S2 x S2. We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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