scholarly journals Two mod-p Johnson filtrations

2015 ◽  
Vol 07 (02) ◽  
pp. 309-343 ◽  
Author(s):  
James Cooper
Keyword(s):  
Free Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Heegaard Splitting ◽  
Mapping Class ◽  
Central Series ◽  
Rational Homology ◽  
Definition Of ◽  
Mod P

We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group, we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorphisms. We calculate the image of the first of these homomorphisms. We give generators for the kernels of these homomorphisms as well. We restrict the range of our mod-p Johnson homomorphisms using work of Morita. We finally prove the announced result of Perron that a rational homology 3-sphere may be given as a Heegaard splitting with gluing map coming from certain members of our mod-p Johnson filtrations.

scholarly journals The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

2019 ◽  
Vol 11 (02) ◽  
pp. 273-292
Author(s):  
Charalampos Stylianakis
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Mapping Class Group ◽  
Abelian Subgroup ◽  
Infinite Order ◽  
Class Group ◽  
Normal Closure ◽  
Mapping Class ◽  
Infinite Index ◽  
Half Twist

In this paper we show that the normal closure of the [Formula: see text]th power of a half-twist has infinite index in the mapping class group of a punctured sphere if [Formula: see text] is at least five. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the [Formula: see text]th power of a Dehn twist has infinite order, and for some integers [Formula: see text] the quotient contains a free group. As a second corollary we recover a result of Coxeter: the normal closure of the [Formula: see text]th power of a half-twist in the braid group of at least four strands has infinite index. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on [Formula: see text]-graphs, as introduced by Kazhdan–Lusztig.

Product of Two Commutators as a Square in a Free Group

1990 ◽  
Vol 33 (2) ◽  
pp. 190-196
Author(s):  
Jonell A. Comerford ◽  
Y. Lee
Keyword(s):  
Free Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Algebraic Interpretation

AbstractWe show that, if [s,t][u, v] = x2 in a free group, x need not be a commutator. We arrive at our example by use of a result of D. Piollet which characterizes solutions of such equations using an algebraic interpretation of the mapping class group of the corresponding surface.

scholarly journals ABELIAN COVERS OF SURFACES AND THE HOMOLOGY OF THE LEVEL L MAPPING CLASS GROUP

2011 ◽  
Vol 03 (03) ◽  
pp. 265-306 ◽  
Author(s):  
ANDREW PUTMAN
Keyword(s):  
Group Ring ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Homology Group ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Rational Group ◽  
Rational Homology ◽  
Abelian Covers

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian ℤ/L-cover of the surface. If the surface has one marked point, then the answer is ℚτ(L), where τ(L) is the number of positive divisors of L. If the surface instead has one boundary component, then the answer is ℚ. We also perform the same calculation for the level L subgroup of the mapping class group. Set HL = H1(Σg; ℤ/L). If the surface has one marked point, then the answer is ℚ[HL], the rational group ring of HL. If the surface instead has one boundary component, then the answer is ℚ.

scholarly journals MAPPING CLASS GROUPS OF HEEGAARD SPLITTINGS

2013 ◽  
Vol 22 (05) ◽  
pp. 1350018 ◽  
Author(s):  
JESSE JOHNSON ◽  
HYAM RUBINSTEIN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Heegaard Splitting ◽  
Mapping Class ◽  
Connected Components ◽  
Heegaard Splittings ◽  
Ambient Manifold ◽  
Periodic Element ◽  
Heegaard Surface ◽  
Structural Theorems

The mapping class group of a Heegaard splitting is the group of connected components in the set of automorphisms of the ambient manifold that map the Heegaard surface onto itself. We find examples of elements of the mapping class group that are periodic, reducible and pseudo-Anosov on the Heegaard surface, but are isotopy trivial in the ambient manifold. We prove structural theorems about the first two classes, in particular showing that if a periodic element is trivial in the mapping class group of the ambient manifold, then the manifold is not hyperbolic.

scholarly journals Mod p homology of the stable mapping class group

Topology ◽  
2004 ◽  
Vol 43 (5) ◽  
pp. 1105-1132 ◽  
Author(s):  
Søren Galatius
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Stable Mapping ◽  
Mod P

Dehn Twists

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Free Group ◽  
Mapping Class Group ◽  
Infinite Order ◽  
Class Group ◽  
Intersection Number ◽  
Mapping Class ◽  
Algebraic Operation ◽  
Closed Curves ◽  
Dehn Twists ◽  
Rank 2

This chapter deals with Dehn twists, the simplest infinite-order elements of Mod(S). It first defines Dehn twists and proves that they are nontrivial elements of the mapping class group. In particular, it considers the action of Dehn twists on simple closed curves. As one application of this study, the chapter proves that if two simple closed curves in Sɡ have geometric intersection number greater than 1, then the associated Dehn twists generate a free group of rank 2 in Mod(S). It also proves some fundamental facts about Dehn twists and describes the center of the mapping class group, along with algebraic relations that can occur between two Dehn twists. Finally, it explores three geometric operations on a surface that each induces an algebraic operation on the corresponding mapping class group: the inclusion homomorphism, the capping homomorphism, and the cutting homomorphism.

scholarly journals ON THE MAPPING CLASS GROUP OF A HEEGAARD SPLITTING

2013 ◽  
Vol 56 (1) ◽  
pp. 93-101
Author(s):  
CHARALAMPOS CHARITOS ◽  
IOANNIS PAPADOPERAKIS ◽  
GEORGIOS TSAPOGAS
Keyword(s):  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Heegaard Splitting ◽  
Mapping Class ◽  
Group Of Automorphisms

AbstractFor the mapping class group of 3-manifold with respect to a Heegaard splitting, a simplicial complex is constructed such that its group of automorphisms is identified with the mapping class group.

AN EXPANSION OF THE JONES REPRESENTATION OF GENUS 2 AND THE TORELLI GROUP II

2004 ◽  
Vol 13 (02) ◽  
pp. 297-306 ◽  
Author(s):  
YASUSHI KASAHARA
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Algebraic Property ◽  
Linear Representation ◽  
Lower Central Series ◽  
Mapping Class ◽  
Central Series ◽  
Torelli Group ◽  
Genus 2 ◽  
Closed Oriented Surface

We continue to study the algebraic property of the linear representation of the mapping class group of a closed oriented surface of genus 2 constructed by V. F. R. Jones. We consider the perturbation of the representation at the involved parameter t=1. This perturbation naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients. Our arguments follow the same line of our previous paper which dealt with the perturbation at t=-1. However, the obtained results may still suggest a new aspect of the representation.

scholarly journals Nilpotency and triviality of mod p Morita-Mumford classes of mapping class groups of surfaces

2002 ◽  
Vol 165 ◽  
pp. 1-22 ◽  
Author(s):  
Toshiyuki Akita
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Supporting Evidence ◽  
Class Groups ◽  
Oriented Surface ◽  
Closed Oriented Surface ◽  
Mod P

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

scholarly journals HIGH DISTANCE KNOTS IN CLOSED 3-MANIFOLDS

2012 ◽  
Vol 21 (02) ◽  
pp. 1250017 ◽  
Author(s):  
MARION MOORE CAMPISI ◽  
MATT RATHBUN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Heegaard Splitting ◽  
Mapping Class ◽  
Genus 2

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V, W) is a Heegaard splitting of genus ≥2, then the coarse mapping class group of (S, V, W) is isomorphic to the mapping class group of (S, V, W).

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