In this group led discussion, marie will demonstrate mind mapping techniques which can be used from test planning through to test closure activities. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Oneended subgroups of mapping class warwick insite. Mapping class groups of punctured surfaces and maybe. Class group calculations by minkowski bound keith conrad. Definition of mapping class group in the definitions. Mapping class groups are discussed m section ll, with an emphasis on the torelli group. This is a strengthening of the result of hamenst adt and of behrstock, kleiner, minsky and mosher that the mapping class groups are quasiisometrically rigid. Problems on homomorphisms of mapping class groups m. The factors of z correspond to the dehn twists around the boundary circles of the holes.
The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Nov 01, 2016 we say that the mapping class group has an isometric action on the curve graph, so to every element of the group we associate an isometry of the graph, which is a map which preserves distances between elements. Mapping class group factorizations and symplectic 4manifolds. Lectures on the mapping class group of a surface thomas kwokkeung au, feng luo, and tian yang abstract. We will give the complete list of the rst homologies and we calculate them.
The torelli group and representations of mapping class groups. In this paper, we survey some rigidity results on actions of mapping class groups. The mapping class group mcg of s is the group of all isotopy classes of orientation preserving selfhomeomorphisms of s. Every homeomorphism h of b induces a homeomorphism of the boundary s. The torelli group and representations of mapping class groups tara e. Mapping class groups and classifying surface automorphisms joseph wells arizona state university november, 2015 the fundamental group of a topological space is a familiar and important algebraic invariant. The dehnnielsenbaer theorem states that for a closed, connected and orientable surface m the extended mapping class group of m is isomorphic to the outer automorphism group of the fundamental g. This resource provides suggestions on key traits to consider when recruiting small group leaders and allows you to map groups by type, age and gender. Mapping class groups do not have kazhdans property t. We prove that the mapping class group of a closed oriented surface of genus at least two does not have kazhdans property t. Ideal class groups of number fields are fundamental objects in number theory.
Introduction these notes are based on lectures given at zhejiang university, july 1420, 2008. A primer on mapping class groups math the university of utah. On the 4rank of class groups of quadratic number fields algant. Generators for the mapping class group request pdf. Ten problems on the mapping class groups amazon s3. We refer to 16, 10 for a general introduction to mapping class groups and to 11 for a recent survey on dynamical properties of the action of outg on representation varieties repg. Journey mapping is a process that provides a holistic view of the customer experience by uncovering moments of both frustration and delight throughout a series of interactions. Mapping class groups and classifying surface automorphisms. In other words, the mapping class group is the group of homeomorphism of x x onto itself, modulo isotopy. Database class mapping wizard in pega pega knowledge sharing. Now the mapping class group acts on pi well, to be more precise, since a diffeomorphism of the surface might move the basepoint around, you have a homomorphism.
Our book is now available from princeton university press. The mapping class group of m is interpreted as the group of isotopyclasses of automorphisms of m. We refer to i02 for a summary of the basic properties of the mapping class group and for references. The dehnnielsenbaer theorem states that for a closed, connected and orientable surface m the extended mapping class group of m is isomorphic to the outer automorphism group of.
An elementary proof of the second homology of the mapping class group was recently given by the author and stipsicz in 30. Connections with 3manifolds, symplectic geometry and algebraic geometry 129 chapter 9. Pdf it is shown, that the mapping class group of a surface of the genus g 1 admits a faithful representation into the matrix group gl 6g6, z. Oneended subgroups of mapping class groups 3 in fact, we could take gto be any gromov hyperbolic space, though we will deal with a graph here to avoid technical details. In the case where is a closed orientable surface group, theorem 1. Ten problems on the mapping class groups nikolai v. The group of selfhomeomorphisms of a topological space, modulo homotopy.
Abstract we study the large scale geometry of the mapping class group, mcgs. Minsky these notes summarize a short set of lectures i gave at the pcmi program on mapping class groups and moduli spaces in the summer of 2011. These techniques have been around for a while, with origins going back several hundred years. Relation between the braid group and the mapping class group. Information and translations of mapping class group in the most comprehensive dictionary definitions resource on the web. Actions of mapping class groups hal archive ouverte. In these lectures, we give the proofs of two basic theorems on surface topology, namely, the work of dehn and lickorish on generating the mapping class group of a surface by dehntwists. Mind mapping in testing has become popular in recent years as it assists in lateral. The distance between two points in the graph is just the smallest number of edges we need to pass along to get from one to the other. The group is the mapping class group of s, denoted by mods. Pega supports mapping class to the table, so that when we save a page of defined class, the particular instance gets saved to the mapped database table. Minsky dedicated to the memory of candida silveira. This is the same as the mapping class group of the torus minus a point x, which acts on loops based at x.
Remember only concrete classes can be mapped to dedicated database table. Mapping class groups and curves in surfaces picture this maths. Steve humphries showed in 14 that one has the following set of generators for the mapping class group of a surface of genus g. Mapping class group how is mapping class group abbreviated.
Dimension and rank for mapping class groups by jason a. All the analogies and parallels between mapping class groups and other groups show how mathematics is connected. Infinite loop space structures on the stable mapping class. The first statement is due to, see also at sphere eversion. Definition of mapping class group in the dictionary. Examples include heegaard decompositions of 3manifolds, monodromies of surface bundles, and monodromies of lefschetz. These include the relation between the structure of the mapping class group and invariants of 3manifolds, the unstable cohomology of the moduli space of curves. Z gg 0,1 of a disk with g holes the boundary of which are parametrised. It is shown, that the mapping class group of a surface of the genus g 1 admits a faithful representation into the matrix group gl 6g6, z. A linear representation of the mapping class group a and the.
The least possible cardinality of dehn twists generating the mapping class group of a closed surface of genus g. In number theory, the ideal class group or class group of an algebraic number field k is the. Class group calculations keith conrad the minkowski bound says, for a number eld k, that any ideal class contains an integral ideal with norm bounded above by n. Introduction there are at least two interesting actions of the mapping class group modg. The proof is based on a categorical correspondence between the riemann surfaces and the socalled toric afalgebras. If the class number is prime, then of course the class group is. Due to contractual obligations, we will not be posting the published version on the web site. My goal was to present some analytic techniques that can be used to study the action of the mapping class group on the representations varieties of surface groups. Whenever the mapping class group of a closed orientable surface of genus gacts by semisimple isometries on a complete cat0 space of dimension less than git xes a point. From these properties of cut systems it is a routine matter to read off a finite presentation for the mapping class group of m, the group of isotopy classes of.
P consisting of principal divisors that can be generated by a totally positive ele ment. We prove that certain obstructions to the existence of a faithful linear representation do not exist in mg. If x x is a smooth manifold, then the mapping class group is the group of connected components of the diffeomorphism group. This group is also known as the teichmuller modular group of s, whence the notation. So if m is a topological manifold, the mapping class group is the group of isotopyclasses of homeomorphisms of m. Problems on mapping class groups and related topics. Our main result is that for any asymptotic cone of mcgs, the maximal dimension of locally compact subsets coincides with the maximal. In particular, the mapping class group of a surface is a finitely generated group.
A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the literature, can be found in section 2 of a paper of mine with nathalie wahl. Somewhat less familiar, but equally as important, is the mapping class group of a topological space. Along the way, central objects and tools are introduced, such as the birman exact sequence, the complex of curves, the braid group, the. A primer on mapping class groups begins by explaining the main group theoretical properties of mods, from finite generation by dehn twists and lowdimensional homology to the dehnnielsenbaer theorem. The mapping class group, m, of m is thegroup of isotopy classes oforientation preserving. Then pi is a free group endowed with a set s of n distinguished conjugacy classes, namely the conjugacy classes of loops around punctures. In particular, the ideal class group is generated by the prime ideals with norm not exceeding this bound. Most often it is used in the context of a manifold m. Equivalently it is the group of connected components of aut x \mathbfautx. As a developer all we need to do is, create a new concrete class. We will prove the ideal class group of every number. We will study these two objects and how they relate to each other.
First, we use the notation in figure 9 with the notation introduced. A brief introduction to mapping class groups yair n. The mapping class groups of all compact orientable 3manifolds are essentially known. Reflection principles and bounds for class group torsion. Ivanov 19941998, 2014 let sbe a compact orientable surface, possibly with a boundary. For example, mccarthy and ivanov independently proved that mapping class groups satisfy a version of the tits alternative. Mapping class group factorizations and symplectic 4manifolds 3 gompf,theargumentextendstothecaseoflefschetz.
We say that the mapping class group has an isometric action on the curve graph, so to every element of the group we associate an isometry of the graph, which is a map which preserves distances between elements. My job was to introduce the mapping class group of a surface, discuss its basic features from a. A primer on mapping class groups begins by explaining the main grouptheoretical properties of mods, from finite generation by dehn twists and lowdimensional homology to the dehnnielsenbaer theorem. B of b and we get an embedding of mg into the mapping class group mcgs of the surface s. Lowdimensional homology groups of mapping class groups. We refer to i02 for a summary of the basic properties of the. The second and third are due to earleeells 67, gramain 73. Done successfully, it reveals opportunities to address customers pain points, alleviate fragmentation, and, ultimately, create a better experience for your users. However, many linear group results inspire mapping class group analogues. Stabilization for mapping class groups of 3manifolds, duke math. The computation of class numbers, rather than class groups, can be obtained by analytic methods. We shall elaborate on arguments from ba for a proof in the general. By creating influence network maps, individuals and groups can clarify their own view of a situation, foster discussion, and develop a strategic approach to their networking activities. The braid group is trivial, is an infinite cyclic group, and is isomorphic to the knot group of the trefoil knot in particular, it is an infinite nonabelian group.
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