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Results tagged with gt.geometric-topology
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user 39654
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).
2
votes
Geometry and topology of Fuchsian character varieties
Let $\Gamma< PSL(2,\mathbb R)$ (not $SL(2,\mathbb R$)!) be a cocompact Fuchsian group. Given a Lie group $G$ one defines
The representation variety $R(\Gamma, G)=Hom(\Gamma,G)$. One further defines $ …
- 12.3k
8
votes
Accepted
Ergodicity of action of finite index subgroups in the boundary
Let $X$ be a Riemann surface of class $P_G$ (i.e. which carries a Green function) but is Liouville (i.e. admits no nonconstant bounded harmonic functions). One way to construct these is to take a $\m …
- 12.3k
5
votes
Accepted
Is a simply connected locally 2-connected complex a union of spheres and planes?
I checked: Bing's "house with two rooms" (see for instance here for a nice picture) is an example: It is a contractible but not collapsible finite 2-dimensional complex. Since it is contractible, it c …
- 12.3k
17
votes
Is there a continuous partition of space into circles?
Yes, there is a topological foliation of $\mathbb R^3$ by smooth circles. A foliation by topological circles was constructed in by Vogt in
Vogt, Elmar, A foliation of ${\mathbb{R}}^3$ and other punctu …
- 12.3k
5
votes
Accepted
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Expanding on my comments, here are some obstructions coming from Hausdorff dimension and self-similarity.
One observation is that every nonelementary Kleinian group $\Gamma$ has positive critical exp …
- 12.3k
5
votes
Cohomological gap in arithmetic groups
This is to record our current lack of understanding of cohomological dimensions of subgroups of arithmetic groups. I will limit myself to the case of uniform lattices since the situation with nonunifo …
- 12.3k
3
votes
Accepted
Representation of Lie groups inducing a quasi-isometric embedding of their symmetric spaces
Karpelevich and Mostow independently proved that there exists an equivariant totally-geodesic embedding $f: X_1\to X_2$ (irreducibility is irrelevant, all you need is compactness of the kernel). By e …
- 12.3k
5
votes
Accepted
Lengths of generators of surface group
In order to remove this question from the "unanswered list." Let $\epsilon>0$ be the Margulis constant for the hyperbolic plane (with curvature $-1$). Then for every complete hyperbolic surface $S$, i …
- 12.3k
2
votes
Accepted
Uniqueness of a properly convex projective domain divisible by a group
Take a discrete rank $n$ free abelian subgroup $\Gamma$ of the group of diagonal matrices in $SL(n+1, \mathbb R)$ with positive diagonal entries. The group $\Gamma$ will preserve each open coordinate …
- 12.3k
5
votes
Accepted
Residual finiteness and a gluing problem
Thurston never finished his project, hence, we cannot know for sure what exactly did he have in mind in this part of the diagram. Here is what we know:
Fundamental groups of good compact 3-dimensiona …
- 12.3k
22
votes
When does a group act effectively and holomorphically on some Riemann surface?
Donu's answer is correct but amounts to killing a fly with a gun shot: Greenberg proves a harder result than the one needed for the problem.
Theorem. Let $G$ be a countable group. Then there exists a …
- 12.3k
2
votes
Given an embedded disk in $\mathbb{R}^n$, is there always another disk which intersects it n...
Here is a proof in the 2-dimensional case. It does not generalize in higher dimensions and more ideas would be needed. Reading R.Berlanga "A mapping theorem for topological sigma-compact manifolds", C …
- 12.3k
1
vote
Accepted
Inheritance of arithmeticity properties in orbifold strata
Here is what I think is the correct setup:
Let $X$ be a symmetric space of noncompact type, $\Gamma$ is a lattice in the isometry group of $X$. Then $\Gamma$ has finitely many $\Gamma$-conjugacy class …
- 12.3k
12
votes
Accepted
Extending diffeomorphisms
The answer is positive and follows from Corollary 2 in
Palais, Richard S., Extending diffeomorphisms, Proc. Am. Math. Soc. 11, 274-277 (1960). ZBL0095.16502.
(A caveat: Palais is not entirely clear ab …
- 12.3k
4
votes
Accepted
Books for learning branched coverings
Montesinos wrote several papers defining the meaning of branched coverings and proving basic properties(not just between manifolds, but for general topological spaces):
Montesinos-Amilibia, José María …
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