MAPPING CLASS GROUPS OF HEEGAARD SPLITTINGS

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.

Download Full-text

Two mod-p Johnson filtrations

Journal of Topology and Analysis ◽

10.1142/s1793525315500120 ◽

2015 ◽

Vol 07 (02) ◽

pp. 309-343 ◽

Cited By ~ 4

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.

Download Full-text

ON THE MAPPING CLASS GROUP OF A HEEGAARD SPLITTING

Glasgow Mathematical Journal ◽

10.1017/s0017089513000104 ◽

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.

Download Full-text

HIGH DISTANCE KNOTS IN CLOSED 3-MANIFOLDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511009637 ◽

2012 ◽

Vol 21 (02) ◽

pp. 1250017 ◽

Cited By ~ 3

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).

Download Full-text

Generating the Mapping Class Group

10.23943/princeton/9780691147949.003.0005 ◽

2017 ◽

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Exact Sequence ◽

Simplicial Complex ◽

Mapping Class Group ◽

Class Group ◽

Mapping Class ◽

Detailed Structure ◽

Inductive Step ◽

Closed Curves ◽

Combinatorial Object ◽

Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

Download Full-text

The local-to-global property for Morse quasi-geodesics

Mathematische Zeitschrift ◽

10.1007/s00209-021-02811-w ◽

2021 ◽

Author(s):

Jacob Russell ◽

Davide Spriano ◽

Hung Cong Tran

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Type Theorem ◽

Hyperbolic Groups ◽

Global Property ◽

Mapping Class ◽

Relatively Hyperbolic Groups ◽

Convex Cocompact ◽

Relatively Hyperbolic ◽

Translation Length

AbstractWe show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.

Download Full-text

Finiteness for mapping class group representations from twisted Dijkgraaf–Witten theory

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216518500438 ◽

2018 ◽

Vol 27 (06) ◽

pp. 1850043 ◽

Cited By ~ 1

Author(s):

Paul P. Gustafson

Keyword(s):

Braid Group ◽

Mapping Class Group ◽

Class Group ◽

Braid Groups ◽

Fusion Category ◽

Mapping Class ◽

Group Representations ◽

Colored Graphs ◽

Finite Image ◽

Spanning Set

We show that any twisted Dijkgraaf–Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image. This generalizes work of Etingof et al. showing that the braid group images are finite [P. Etingof, E. C. Rowell and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008)(1) 33–42]. In particular, our result answers their question regarding finiteness of images of arbitrary mapping class group representations in the affirmative. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface. To do this translation, we use the fact that any twisted Dijkgraaf–Witten representation associated to a finite group [Formula: see text] and 3-cocycle [Formula: see text] is isomorphic to a Turaev–Viro–Barrett–Westbury (TVBW) representation associated to the spherical fusion category [Formula: see text] of twisted [Formula: see text]-graded vector spaces. The representation space for this TVBW representation is canonically isomorphic to a vector space of [Formula: see text]-colored graphs embedded in the surface [A. Kirillov, String-net model of Turaev-Viro invariants, Preprint (2011), arXiv:1106.6033 ]. By analyzing the action of the Birman generators [J. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–242] on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.

Download Full-text

AUTOMORPHISMS OF BRAID GROUPS ON S2, T2, P2 AND THE KLEIN BOTTLE K

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216508005951 ◽

2008 ◽

Vol 17 (01) ◽

pp. 47-53 ◽

Cited By ~ 2

Author(s):

PING ZHANG

Keyword(s):

Braid Group ◽

Mapping Class Group ◽

Class Group ◽

Klein Bottle ◽

Closed Surface ◽

Group Extension ◽

Braid Groups ◽

Mapping Class ◽

Group B ◽

Central Automorphisms

It is shown that for the braid group Bn(M) on a closed surface M of nonnegative Euler characteristic, Out (Bn(M)) is isomorphic to a group extension of the group of central automorphisms of Bn(M) by the extended mapping class group of M, with an explicit and complete description of Aut (Bn(S2)), Aut (Bn(P2)), Out (Bn(S2)) and Out (Bn(P2)).

Download Full-text

ON POWERS OF HALF-TWISTS IN M(0, 2n)

Glasgow Mathematical Journal ◽

10.1017/s0017089517000143 ◽

2017 ◽

Vol 60 (2) ◽

pp. 333-338 ◽

Cited By ~ 1

Author(s):

GREGOR MASBAUM

Keyword(s):

Mapping Class Group ◽

Class Group ◽

Normal Closure ◽

Mapping Class ◽

Infinite Index ◽

Skein Theory ◽

Half Twist

AbstractWe use elementary skein theory to prove a version of a result of Stylianakis (Stylianakis, The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere, arXiv:1511.02912) who showed that under mild restrictions on m and n, the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.

Download Full-text