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Question 1

0votes

0answers

42views

Proof that a mapping set is a vector space

I'm following Serge Lang's Linear Algebra third edition for the definition of a vector space. A set $V$ of objects over a field $F$ which can be added and multiplied by elements of $F$ is a vector ...

William Avila Aguilar

39

asked1 hour ago

Question 2

1vote

1answer

86views

Mapping torus of a surface with monodromy a separating Dehn twist

Let $S$ be a closed, oriented surface of genus $g\geq 2$ and let $\gamma\subset S$ be a separating, essential curve.For example we can consider $S = \mathbb T^2\# \mathbb T^2$ and $\gamma$ being the ...

Overflowian

6,052

askedOct 9 at 16:07

Question 3

2votes

1answer

61views

Horoballs that are disjoint from all simple geodesics

I'm reading Farb and Margalit's Primer, and they claimed thatOne can choose sufficiently small disjoint open horoball neighborhoods of the cusps (of a hyperbolic surface) so that every essential ...

duang

179

askedSep 17 at 18:10

Question 4

2votes

0answers

40views

The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$

This question follows from a former question of mine, whereLee Morsher answered briefly, "The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$ holds for all surfaces. A ...

user1249906

109

askedJul 27 at 4:42

Question 5

5votes

1answer

77views

Punctured and marked mapping class groups

For the generalized Birman exact sequence, on A Primer on Mapping Class Groups, it's given by $$1\to \pi_1(C_n(S,n))\to \text{MCG}(S, P)\to \text{MCG}(S)\to 1,$$ where the group $\text{MCG}(S, P)$ is ...

user1249906

109

askedJul 26 at 15:57

Question 6

1vote

0answers

27views

Mapping class group of $(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$

I am interested in the mapping class group of $(S^1 \times S^{d-1})\sharp (S^1 \times S^{d-1}) $ in various dimensions. If $d=2$, that's the mapping class group of a genus-2 surface and there are many ...

XYX

121

askedJul 14 at 5:55

Question 7

2votes

1answer

87views

When are two pants decompositions related by a mapping class on a surface?

Let us call a multicurve $\gamma$ in a closed surface of genus $g$, $\Sigma_g$, a full system if it $\Sigma_g$ into a collection of punctured spheres. I've heard people saying (and possibly read ...

HEnnes

73

askedJun 26 at 13:02

Question 8

0votes

0answers

19views

Markov partitions of a pseudo-Anosov homeomorphisms are bounded above

In the book A Primer on Mapping Class Groups, beginning of page 445, the writer says thatany pseudo-Anosov mapping class of a fixed surface has a transitionmatrix whose size (number of rows) is ...

xtwxtw

279

askedJun 14 at 6:57

Question 9

1vote

0answers

56views

Is there the correspondence between conjugacy fundamental groups and lifts of a curve? (Farb & Margalit, p. 23)

I am reading 'A Primer on Mapping Class Groups' by Benson Farb and Dan Margalit. Concerning the material on page 23, I'm having difficulty understanding the one-to-one correspondence between the ...

Topo

107

askedMay 27 at 5:54

Question 10

1vote

0answers

45views

Entropy and stretch factor of pseudo-anosov diffeomorphism (reference request)

Let $f$ be a pseudo-anosov diffeomorphism of an $n$-punctured sphere, with $n\geq 4$.I think it should be true that $h_{top}(f) = \ln \lambda_f$, where $\lambda_f$ is the stretch factor, also called &...

unsure

101

askedApr 21 at 17:44

Question 11

5votes

0answers

123views

Why is the center of the Mapping class group of a genus $g \geq 3$ surface trivial

I am studying about the center of the mapping class group of a genus $g$ surface. I am having some difficulties in understanding the proof for the triviality of the group $Z(Mod(S_{g}))$.Here is the ...

Patrick Bateman

662

askedApr 15 at 5:03

Question 12

0votes

0answers

30views

Graph arising due to Alexander Method [duplicate]

I am trying to compute the center of the mapping class group of a surface of genus greater than 3. For that reason we build an alexander system of curves. The alexander system of curves is isomorphic ...

user982931

askedApr 13 at 18:01

Question 13

1vote

0answers

55views

Seeking clarifications in the statement of Alexander's method in Mapping Class Groups

This is the Alexander Method from "The Primer" :Let $S$ be a compact surface, possibly with marked points, and let $\phi \in Homeo^{+}(S, \partial S)$. Let $\gamma_{1}, \dots, \gamma_{n}$ ...

user982931

askedApr 9 at 17:53

Question 14

2votes

0answers

212views

Mapping class group of the four punctured sphere

I am reading about Mapping Class Groups from "The Primer" by Benson Farb and Dan Margalit. I am stuck with the proof for the mapping class group of the four punctured sphere. Here is the ...

Patrick Bateman

662

askedMar 31 at 13:47

Question 15

1vote

1answer

79views

Embedding 4-holed sphere in handlebody

Let $\Sigma_0^4$ be a 2-sphere with 4 boundary components. There is a curve $\gamma \subset \Sigma_0^4$ as in this picture:Now embed $\Sigma_0^4$ inside a genus $g \geq 4$ handlebody $V_g$ in such a ...

Chase

567

askedMar 13 at 15:03

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Stack Exchange Network

Questions tagged [mapping-class-group]

Proof that a mapping set is a vector space

Mapping torus of a surface with monodromy a separating Dehn twist

Horoballs that are disjoint from all simple geodesics

The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$

Punctured and marked mapping class groups

Mapping class group of $(S^1\times S^{d-1})\sharp (S^1\times S^{d-1})$

When are two pants decompositions related by a mapping class on a surface?

Markov partitions of a pseudo-Anosov homeomorphisms are bounded above

Is there the correspondence between conjugacy fundamental groups and lifts of a curve? (Farb & Margalit, p. 23)

Entropy and stretch factor of pseudo-anosov diffeomorphism (reference request)

Why is the center of the Mapping class group of a genus $g \geq 3$ surface trivial

Graph arising due to Alexander Method [duplicate]

Seeking clarifications in the statement of Alexander's method in Mapping Class Groups

Mapping class group of the four punctured sphere

Embedding 4-holed sphere in handlebody

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