Tagged Questions

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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. ...
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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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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 ...
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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 ...
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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 $(ab)^6$ ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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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How to detect a simple closed curve from the element in the fundamental group?

(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be ...
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Convexity of a minimum function

I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following ...
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Iterated Lefschetz numbers

Given a pseudo Anosov mapping class $f:S_{g,n}\rightarrow S_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$? The Lefschetz number of a mapping class $f$ can ...
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Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
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Reference request: Spin structures on surfaces and the spin mapping class group

I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group. What is known about generating the spin mapping class group? Has anybody ...
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The action of torsion of $MCG(S)$ on curve complex

Hi everyone. Let $S$ be a closed surface with genus at least 3, $\alpha, \beta$ be the two vertices of curve complex of $S$ such that $d_{\mathcal {C}(S)}(\alpha, \beta)\geq 3$. My question is ...
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The action of periodic map on the complex of curves

Hi, everyone. Assume $S$ is a genus at least 2 orientable closed surface. And there is a simplical complex defined on $S$ called Curve complex. It is well known that any automorphism of surface ...
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Eilenberg-Mac Lane spaces for surface group extensions.

(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.) ...
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The image of the point-pushing group in the hyperelliptic representation of the braid group

Let $B_{2g+1}$ be the Artin braid group on $2g+1$ strands. There is a symplectic representation $\rho: B_{2g+1} \rightarrow Sp_{2g}(\mathbf{Z})$ called the "hyperelliptic representation," which ...
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Do the following set of Dehn twists generate the mapping class group?

If $S$ is the surface illustrated below, do the Dehn twists about the red curves generate the mapping class group $\operatorname{MCG}(S,\partial S)$?
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Requiring references

Assume $V$ be a genus larger than 1 handlebody, $S=\partial_{+} V$. Denote $N$ be the normal closure of $MCG(V)$ in $MCG(S)$. Is there any material related to the quotient group $MCG(S)/N$ ? Thanks! ...
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Multiple Dehn twists and minimal position

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit. Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points ...
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Lower bounds on dimensions of faithful representations of braid groups

Let $B_n$ be the braid group on $n$ strands. It's a theorem of Daan Krammer and Stephen Bigelow that there is a a faithful representation $$B_n \to GL_{n \choose 2} \mathbb Z[t^{\pm}, q^{\pm}]$$ ...
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Nielsen-Thurston classification via the curve complex?

I am curious to see if anyone knows a proof of the Nielsen-Thurston classification of mapping classes that does not depend on results in Teichmuller theory. From a naive point of view, translation ...
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Flips of triangulations on non-orientable surfaces

Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures. I'm looking at ideal triangulations of the surface, namely ...
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Automorphisms of Riemann surface and mapping class

For a higher genus Riemann surface $\Sigma$, is it true that every nontrivial (holomorphic) automorphism is of nontrivial mapping class, i.e., not isotopic to the identity?
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Elegant proof that mapping class groups are generated by Dehn twists?

One of the foundational results about mapping class groups of surfaces is that they are generated by Dehn twists. A mapping class is a connected component in a space of diffeomorphisms, so another way ...
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The signature of a mapping torus

Consider a manifold $M$ of dimension $4k + 2$, $k$ an integer. Pick a diffeomorphism $\phi$ of $M$ and construct the mapping torus $T$ of $\phi$. Suppose that there is a $4k+4$ dimensional manifold ...
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Splitting lemma for non abelian groups and mapping class group

Hi everybody! I'm in trouble with a version of the splitting lemma in the non abelian case. To be more specific, I'm working with mapping class groups of surfaces, i.e. the groups of isotopy classes ...
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Realizing braid group by homeomorphisms

Markovich and Saric proved the following remarkable theorem. Let $S$ be a compact surface of genus at least $2$ and let $MCG(S)=\pi_0(Homeo^{+}(S))$ be the mapping class group of $S$. There is then ...
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Is the Action of the mapping class group transitive on embedded arcs?

Let S be a surface of genus g with some parked points (n of them). Assume $n \geq 2$ and fix two of the marked points. Consider the set of embedded arcs going between these two special points. The ...
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pseudo-Anosov maps on surfaces with boundary

In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed ...
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A question in R.C.Penner's paper about Teichmuller space

In R.C.Penner "Decorated Teichmuller theory of boarded surface", on Page 7 and 8, it says that (without proof) the Teichmuller space of surface with $s$ labelled punctures and $r$ labelled boundary ...
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Interesting representations/cohomology of surface groups?

For purposes of my own, I'm interested in constructing connected spaces, without recourse to geometric realisation or the like, that have non-trivial homotopy groups in dimension 1 and 2 and are not ...
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About the proof of Wajnryb's finite presentation of Mod(S)

I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't ...
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The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
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classification of torus bundles

Assume $M$ a topological space,$f\in Homeo (M)$ then torus bundle $M_f=M\times I/\{(x,0)\sim (f(x),1)|x\in M\}$ obiviouly $f$ plays a significant role in determing the the torus bundle. hence there ...
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Are there results about the group of homeomorphisms of $(T^2-\{*,*\})$ up to isotopy?

I am studying a fiber bundle over circle with fiber $T^2-\{*,*\}$. Since this is a mapping torus, the group $Homeo(T^2-\{*,*\})/isotopy$ plays an important role. Are there some existing theorems on ...
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Mapping Class Groups of Punctured Surfaces (and maybe Billiards)

Where can I find a concrete description of mapping class group of surfaces? I know the mapping class group of the torus is $SL(2, \mathbb{Z})$. Perhaps, there is a simple description for the sphere ...
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Reducible 3d torus bundles

Here reducible means that the mapping class for the fiber is a reducible auto-homeomorph in the sense of Nielsen-Thruston. So, could anyone give me a hint to classify them? In contrast, do you agree ...
Please, any information on the periodic mapping classes of the genus two orientable surface, $O_2$, will be greatly thanked. We had been studying the topological structure of 3d surface bundles and ...
Let $X$ be a compact non-orientable surface, maybe with boundary, and let $\tilde X$ be the orienting cover of $X$. If I understand correctly, any smooth automorphism of $X$ lifts naturally to an ...