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Results tagged with gt.geometric-topology
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user 20787
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
5
votes
pseudo-Anosovs with given action on homology
Here's an outline of a proof. Consider a pseudo-Anosov mapping class $\phi$ such that the infinite cyclic group $C = \langle \phi \rangle$ is malnormal. Using the methods of Ivanov and/or McCarthy one …
- 15.4k
7
votes
Translation distance in the curve complex
In the case that $\psi$ is pseudo-Anosov, the best one can do in general, as far as I know, is to get upper and lower bounds which are linear in translation length. These come from train track conside …
- 15.4k
6
votes
Accepted
projective structure and holonomy
There are two kinds of holonomy, which are well contrasted with each other in the opening paragraphs of the link given.
The first kind is exemplified by the holonomy of a Riemannian metric: parallel …
- 15.4k
11
votes
Can we determine which monodromy of surface gives a fibered knot?
Form the mapping torus $M$, check whether $H_1(M;\mathbb{Z})=\mathbb{Z}$ and is generated by a loop on the torus boundary. If not, it isn't a fibered knot complement. If so, do Dehn filling to produce …
- 15.4k
4
votes
Accepted
Some questions on partial pseudo anosov maps
In answering all of your questions I am going to assume that $S$ has a hyperbolic structure, and that the stable and unstable laminations are geodesic laminations. This point of view is explained in t …
- 15.4k
12
votes
Accepted
Regarding the Thurston norm and the ways that a three-manifold can fiber over the circle
The answer is: yes if the rank of $H_2(M;\mathbb{Z})$ is $\ge 2$; and no if the rank is $1$ because in that case there is up to isotopy a unique connected surface bundle structure on $M$. The proof us …
- 15.4k
8
votes
Accepted
Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus
Take any pair of measured laminations $\lambda,\mu$ which each fill the surface and are transverse to each other. Take sequences $\sigma_i,\tau_i$ in Teichmuller space, such that $\sigma_i$ converges …
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7
votes
Accepted
In which cases a fiber bundle over a circle is a graph-manifold?
$M$ is a graph manifold if and only if $\phi$ is not pseudo-Anosov and, in the reducible case, no $\phi$-orbit of components of the complete reduction of $\phi$ is pseudo-Anosov.
To prove this cut $ …
- 15.4k
4
votes
Accepted
Conditions for a graph to be the 1- skeleton of a Surface
It suffices to consider a connected graph. Start from a point, which is the 1-skeleton of a sphere. By induction, consider a connected graph $G$ and an edge $E$, and let $S$ be the surface in which th …
- 15.4k
13
votes
Compelling evidence that two basepoints are better than one
In my proof that mapping class groups are automatic, Ann. of Math. (2) 142 (1995), no. 2, 303–384, I used a theorem from ECHLPT "Word Processing in Groups" which says that if a groupoid is automatic …
- 15.4k
4
votes
Hyperbolic pair of pants.
No it does not. Suppose that the three boundary components have equal and very long length $R$. Then the pair of pants is almost isometric to a graph having two vertices $V,W$ and three edges of lengt …
- 15.4k
8
votes
Accepted
interval exchange maps and surfaces
Your method certainly works, because you are just identifying boundary edges of the annulus $[0,1] \times S^1$ in pairs to form a surface. As usual, when one glues up edge pairs of a surface-with-boun …
- 15.4k
8
votes
A simple closed curve on a surface
Check out the book Thurston's Work on Surfaces for a treatment of Dehn-Thurston coordinates which is simultaneously intuitive and in-depth.
- 15.4k
6
votes
Accepted
Requiring references
The normal closure of $MCG(V)$ in $MCG(S)$ is all of $MCG(S)$. To see why, we know that $MCG(S)$ is generated by Dehn twists, so it suffices to prove that each Dehn twist is in the normal closure. Con …
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3
votes
Accepted
Is transverse measure on a foliation without closed leaves unique?
Such foliations were studied rather intensely in the early works on measured foliations that introduced them to the mathematical world. See for example "Thurston's work on surfaces" aka "Travaux de Th …
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