All Questions
Tagged with mg.metric-geometry gt.geometric-topology
241 questions
7
votes
0
answers
154
views
Connectedness of cones in the boundary of a 1-ended hyperbolic group
Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
- 611
1
vote
0
answers
95
views
Existence of polytope
Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...
- 43
8
votes
0
answers
185
views
Sharp isoperimetry in the discrete Heisenberg group
The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...
- 4,432
1
vote
0
answers
83
views
Thomsen Blaschke condition
I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof.
Paper 1, page 1, line 10 says : Consider the topological image G of a 2-...
- 221
14
votes
2
answers
786
views
For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?
Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...
- 1,069
8
votes
2
answers
328
views
Equivalence of definitions of quasiconformal surfaces?
I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of quasiconformal surface.
Definition: A quasiconformal surface $S$ is a ...
- 298
15
votes
1
answer
413
views
bi-Lipschitz gluing
Let $H$ be the Heisenberg group with
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.
I have a bi-Lipschitz ...
- 45k
1
vote
2
answers
310
views
Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?
Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?
- 2,964
3
votes
0
answers
383
views
Discontinuity of Radon-Nikodym derivative for Patterson-Sullivan measures for word metrics on Gromov hyperbolic groups
Let $\Gamma$ be a Gromov hyperbolic group coming endowed with a word metric coming from some finite generating set. Let $\nu$ be an associated Patterson-Sullivan measure (quasi-conformal density).
I ...
- 2,964
6
votes
0
answers
383
views
When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?
Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...
- 2,964
8
votes
1
answer
696
views
Geodesics on manifolds with boundary
Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...
- 131
15
votes
1
answer
350
views
Closed geodesics avoiding points in hyperbolic surfaces
Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...
- 380
7
votes
2
answers
355
views
Convex subcomplexes of CAT(0) cubical complexes
Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...
- 8,493
10
votes
1
answer
346
views
A forked plane continuum
I came up with this question while trying to solve the following MO one:
Does every connected set that is not a line segment cross some dyadic square?
Suppose $C$ is a plane continuum (i.e. a ...
- 1,375
5
votes
0
answers
275
views
Is there a connection between |roots| $\rightarrow$ 1 and Gromov's waist theorem?
Recent questions showed that roots of a random polynomial tend to lie on the
unit circle ("Why do roots of polynomials tend to have absolute value close to 1?"; "Distribution of roots of complex ...
- 151k
10
votes
1
answer
1k
views
CAT(0) groups that does not act on CAT(0) cubical complex
CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
- 1,598
6
votes
2
answers
168
views
Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?
For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
- 13.4k
2
votes
1
answer
118
views
Characterization of the medial axis of a surface
I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...
- 163
3
votes
2
answers
240
views
Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?
Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
- 53
4
votes
0
answers
179
views
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: ...
- 380
1
vote
0
answers
74
views
Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometry
Let $M$ be an open Riemannian surface of bounded geometry. Let $\Gamma$ be a group of diffeomorphisms of $M$. Suppose that $\Gamma$ is equi-quasi-isometric; i.e., its elements are (differentiable) ...
- 329
6
votes
2
answers
729
views
Rationality of translation lengths in hyperbolic groups
Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set.
It is a theorem of Gromov ...
- 619
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
8
votes
1
answer
704
views
Is this knot invariant already treated somewhere in the literature?
Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$
We can turn $Y_K$ into a metric space by considering the distance induced by ...
- 197
1
vote
0
answers
156
views
Entangled helical knots
Consider a pair of disjoint, congruent helices $H_1$ and $H_2$
passing through one another in the following sense.
(Caveat lector: This question is not of general interest! It is also long.)
$H_1$ is ...
- 151k
12
votes
1
answer
504
views
Tverberg's theorem in CAT(0) spaces
Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$:
Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
- 31.2k
12
votes
1
answer
375
views
Why do convex polytope options constrict with dimension, rather than expand?
There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...
- 151k
16
votes
1
answer
1k
views
Does Gromov's Waist Inequality imply Borsuk-Ulam?
I'm curious if anyone can see a route to get the Borsuk-Ulam theorem from Gromov's waist inequality. For the sake of notation, here's the inequality:
Let $S^n$ denote the round unit sphere in $\...
- 647
6
votes
2
answers
323
views
Hilbert space compression of the lamplighter group
What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and ...
- 2,449
16
votes
1
answer
1k
views
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 ...
- 828
12
votes
3
answers
371
views
Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?
The geometrization theorem tells us:
Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a complete ...
- 647
3
votes
1
answer
266
views
Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?
Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$.
If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial X$,...
- 10.4k
11
votes
0
answers
352
views
Right-angled polytopes
%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...
- 1,268
7
votes
1
answer
559
views
Standard (special) spines and hyperbolic structure on 3-manifolds
My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
- 367
14
votes
1
answer
3k
views
How metric is Riemannian geometry
Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, dt\,,$$...
- 5,824
2
votes
0
answers
163
views
Reachability in dynamic random graphs
There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...
- 35
13
votes
3
answers
835
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
- 15.5k
0
votes
1
answer
1k
views
Proper Group action on a metric space
Let $(X,d)$ be a metric space and $C\subset X$ be a compact subset. Let furthermore $G$ be a group that acts on $X$ proper and by isometries. Does there exist an $\epsilon >0 $ such that: Let $U=$ {...
- 1
3
votes
1
answer
469
views
simplicial complex equipped with barycenric metric is complete [closed]
Consider a simplicial complex $C$. On its support $$|C|=\lbrace \alpha = \sum_{v\in C}\alpha_{v}v \mid 0\leq \alpha_{v} \leq 1 , \sum_{v\in C}\alpha_{v} =1\mbox{ and }v|{\alpha_{v}} \neq 0\mbox{ is a ...
- 31
15
votes
2
answers
1k
views
Is every connected metrizable locally path connected space a length space?
Does every connected metrizable locally path connected topological space $X$ admit a compatible metric $d$ so that $(X,d)$ is a length space?
(Edit to correct definition: Recall that a metric space $(...
- 1,968
9
votes
0
answers
331
views
Is the connected sum of knots an isometry?
Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map $\...
- 91
6
votes
0
answers
389
views
A conjecture of Thurston and possibly Weeks too
What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
- 1,131
1
vote
1
answer
185
views
Fractal dimension of 1D set, what if logN vs log(e) is a polygonal chain?
I have a finite set of points, and plot the graph log(N) vs. log(e). I see a polygonal chain (the final slope, starting at some size of e, is zero, of course). If the set represents some physical ...
- 93
3
votes
0
answers
239
views
Computing the Volume of Closed 3-Manifolds and the Geometrization Conjecture
My question is whether or not if I generalize Theorem 2(i) of "Contact Graphs of Unit Sphere Packings Revisited" [2012] by K. Bezdek and S. Reid (arXiv link) which states
The number of touching ...
- 1,431
17
votes
0
answers
731
views
Does every connected set that is not a line segment cross some dyadic square?
A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...
- 171
5
votes
1
answer
726
views
QVH characterization of virtually special groups
Agol's recent VHC paper gave a characterization of virtually special groups in terms of being $\mathcal{QVH}$. He remarks that this may be taken as the defining property of virtually special groups ...
- 197
1
vote
1
answer
214
views
Orbits of Product Lie Groups Action
Hi to all,
Let $G$ be a Lie group of linear isometries of $\mathbb{R}^n_{\nu}$ ($\mathbb{R}^n_{\nu}$ is the semi-Euclidean space) and $G_1$ ,$G_2$ two Lie subgroups of $G$. Let $G_1 \times G_2$ as ...
user30435
12
votes
1
answer
409
views
Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite
I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their ...
- 197
2
votes
2
answers
464
views
Combination theorems for discrete subgroups of isometry groups
Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
- 10.4k
17
votes
1
answer
526
views
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...
- 151k