Topology of groups of automorphisms of surfaces, and high dimensional analogues.

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11
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3answers
221 views

Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

I've read that if $M_1, \dots, M_n$ are matrices in $\mathrm{SL}(2, \mathbb{Z})$ whose product is the identity, and each is conjugate to the shear $$ \begin{pmatrix} 1 & 1 \\ 0 & ...
9
votes
1answer
251 views

Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ...
4
votes
1answer
143 views

mapping class group of a two-holed torus

It is well-known that in the picture below we have $t_d=(t_at_b)^6$ as elements in the mapping class group of a two-holed torus, ($t_\gamma$ represents positive Dehn twist about the curve $\gamma$). ...
7
votes
1answer
165 views

Subgroups of the mapping class group of a surface generated by Dehn twists

Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a ...
5
votes
0answers
101 views

Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$. It is known ...
3
votes
1answer
134 views

mapping class group relations

The question I want to ask is vague in a sense. We have examples of mapping class relations, e.g. lantern relation, chain relations, etc. For instance the latern relation on a disk with three boundary ...
4
votes
1answer
137 views

Mapping class group of a punctured genus 0 surface

Let $T_{0,n}$ be the Teichmuller space of $n$-punctured genus $0$ Riemann surface, and $M_{0,n}$ the Moduli space (assume $n\geq 3$ and the punctures are numbered). What is the correct notion of the ...
2
votes
0answers
76 views

cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any ...
7
votes
2answers
236 views

Automorphism of genus 2 surface with 5 fixed points

Is there a self-homeomorphism of a genus 2 (closed, orientable) surface, which has finite order and exactly 5 fixed points? Of course, the same question can be asked replacing 2 by $g$ and $5$ by any ...
0
votes
0answers
94 views

Mapping theorem in higher dimensions

The Riemann mapping theorem states that given any two simply connected open domains $A$ and $B$ of $\mathbb C$ that are neither empty nor equal to $\mathbb C$, there exists a unique (up to ...
3
votes
1answer
81 views

Symmetry conjecture for minimal dilatation pseudo Anosov mapping classes

The conjecture is something like the following: The minimal dilatation among pseudo-Anosov mapping classes on a surface $S_{g,n}$ is realized by $\rho\circ\omega$ where $\omega$ is supported on a ...
3
votes
0answers
138 views

Injectivity of the Dehn-Nielsen-Baer map?

If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class ...
3
votes
1answer
169 views

Defining relations of mapping class group for genus 2 closed surface

We know that mapping class group (MCG) $\Gamma_1$ for genus 1 closed surface is generated by two elements: $U$ of order 6 and $S$ of order 4. There is a defining relation that totally fixed the MCG ...
2
votes
0answers
126 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i ...
5
votes
1answer
269 views

Ivanov's metaconjecture on surface homeomorphisms.

In Fifteen problems about MCG Ivanov stated the following metaconjecture: Every object naturally associated to a surface S and having a sufficiently rich structure has $Mod(S)$ as its groups of ...
2
votes
0answers
86 views

Self-diffeomorphisms of fibered knot complements

A knot $K \subset S^3$ is fibered if the complement $S^3 \setminus K$ of (a small open neighborhood of) $K$ is a fiber bundle over $S^1$. (The fiber will be a surface with one boundary component.) ...
2
votes
1answer
182 views

Torsion elements in the mapping class group

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms ...
4
votes
2answers
227 views

Geometrisation of inclusion-like epimorphisms to free groups

Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$. Call $F_g$ be the free group of rank $g$. Denote by $i:S_g \to ...
1
vote
1answer
348 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
0
votes
1answer
198 views

Action of the pure braid group on the commutator subgroup of a free group

Let $P=P_n$ be the pure braid group on $n$ strands and $F=F_n$ the free group on $n$ generators. I'm interested in a nice description of the action of $P$ on the derived subgroup $F'$ which somehow ...
3
votes
1answer
159 views

Dehn twist about an arbitrary curve

I need to know if there is an algortihm to write down a Dehn twist about an arbitrary curve on an orientable surface $S$ , as product of a set of generators of $MCG(S)$. Since we have the conjugacy ...
3
votes
1answer
72 views

Extension of the group $Sp(2g,\mathbb{Z})$

It is known that there is a surjective homomorphism from the genus $g$ mapping class group $M_g$ to $Sp(2g,\mathbb{Z})$. Is there any non-trivial extension of $Sp(2g,\mathbb{Z})$ by $\mathbb{Z}_2$ , ...
18
votes
3answers
690 views

Which mapping class group representations come from algebraic geometry?

Let $\Gamma_g$ be the mapping class group of a closed oriented surface $\Sigma$ of genus $g$. There is a natural surjection $t \colon \Gamma_g \to \mathrm{Sp}(2g,\mathbf Z)$ which sends a mapping ...
4
votes
1answer
173 views

Classification of elements in mapping class groups

Recently I start learning mapping class group. The Nielsen-Thurston classification says that each element in mapping class group $Mod(S_{g,n}),g,n\geq 0$ is periodic, reducible, or pseudo-Anosov. Take ...
1
vote
0answers
121 views

Mapping class group and fundamental group of a 3 manifold

If the 3 manifold is Haken, then the natural homomorphism from the mapping class group of this 3 manifold to the outer automorphism of its fundamental group is an isomorphism. Any other kind of 3 ...
12
votes
1answer
351 views

Finite subgroups of mapping class groups

Given a closed, oriented surface $\Sigma$ of genus greater than 1, let $Mod(\Sigma)$ denote the mapping class group of orientation preserving diffeomorphisms of $\Sigma$ up to isotopy. Given any ...
3
votes
1answer
185 views

A query about Hatcher flow on arc complex

In the paper "Triangulations of Surfaces" Hatcher proved that the arc complex associated to a punctured surface is contractible. The main proof is divided into two parts. In the first part he assumes ...
6
votes
0answers
331 views

Some questions about geodesic lamination

I'm learning geodesic laminations on surfaces. Here are some questions I thought a lot but could not understand well. We consider a complete finite area hyperbolic surface $S$ w/o geodesic boundary. ...
3
votes
1answer
86 views

Distorsion of subgroups of the mapping class group

Let $S_{g,b}$ be an oriented surface with $b$ boundary components and $S_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S_{g,b})$ and $\mathrm{Mod}(S_g^b)$ their (orientation ...
4
votes
0answers
114 views

Centralizers and intersections in the Gromov-boundary of the mapping class group

The mapping class group of a punctured surface $\Sigma$ is weakly relatively hyperbolic (see below), hence it is well defined the Gromov-boundary with respect to the relative metric. First question: ...
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0answers
46 views

Connectivity and contarctibility of complexes associated to curves and arcs

There are various complexes associated to a surface using the curves and arcs e.g. Curve complex, Arc complex, curve arc complex and so on (for a collection of such objects see This). Now to ...
0
votes
1answer
136 views

A doubt from “Geometry of the complex of curves II: Hierarchical structure” by Masur and Minsky

In the paper "Geometry of the complex of curves II: Hierarchical structure" (Paper) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain ...
3
votes
1answer
151 views

In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In Meyer, Werner Die Signatur von ...
3
votes
2answers
285 views

Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
5
votes
0answers
187 views

Third cohomology of mapping class group

I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...
3
votes
0answers
131 views

Lie Group Isomorphisms

I'm not sure of the difficulty of the question I'm about to ask. If it does not fit the criteria for this site then I apologize in advance, I'm rather new here. So here it goes: Let $G$ be a Lie ...
0
votes
1answer
260 views

from Dehn twists to surgery diagram [closed]

Assume the relation $(ab)^6=1$, for $a$ and $b$ Dehn twists about the meridian and the longitude of a torus. Now if we glue the two ends of $T\times I$ together by either the diffeomorphism $(ab)^6$ ...
4
votes
1answer
151 views

(Un)distorted subgroups in the mapping class group: reference required.

Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results: the cyclic subgroup generated by a pseudo-Anosov element ...
10
votes
1answer
185 views

Mapping class group vs automorphism group in cobordism category

Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms. By sending a diffeomorphism $\phi$ of a surface $X$ to its associated ...
2
votes
2answers
321 views

Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface. The first paper I am aware of that uses the ...
7
votes
1answer
174 views

What is the order of the isotopy group of the Brieskorn homology 3-sphere?

Let $\Sigma(p,q,r)$ be the Brieskorn homology 3-sphere with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (so not the 3-sphere or the Poincare sphere). The fundamental group is given by $$ ...
6
votes
2answers
209 views

Are there some list of the finite subgroups of the mapping class groups of low genus surfaces?

We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, ...
1
vote
1answer
408 views

Explicit computation of the action of a Dehn twist on the fundamental group of a surface

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...
3
votes
2answers
316 views

Action of Mapping Class Group on Arc complex

Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
-1
votes
1answer
158 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
9
votes
1answer
291 views

Mapping class groups of small Seifert-fibred 3-manifolds

Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location? For most Seifert manifolds working out the mapping class group is easy-enough ...
1
vote
1answer
130 views

Centralizer of a pseudo-Anosov element

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?
13
votes
1answer
450 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
3
votes
0answers
184 views

Mapping One Curve to another using Dehn Twists

Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...
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0answers
79 views

Isomorphism type of mapping class group

Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$. Assume that it is not trivial. Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ ...