Questions tagged [mapping-class-group]
For questions related to mapping class group. The mapping class group is a certain discrete group corresponding to symmetries of the space.
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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 ...
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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 ...
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Horoballs that are disjoint from all simple geodesics
I'm reading Farb and Margalit's Primer, and they claimed that
One can choose sufficiently small disjoint open horoball neighborhoods of the cusps (of a hyperbolic surface) so that every essential ...
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The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$
This question follows from a former question of mine, where
Lee Morsher answered briefly, "The isomorphism between $\text{MCG}(S,P)$ and $\text{MCG}(S \setminus \{P\})$ holds for all surfaces. A ...
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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 ...
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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 ...
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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 ...
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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 that
any pseudo-Anosov mapping class of a fixed surface has a transition
matrix whose size (number of rows) is ...
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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 ...
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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 &...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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