# Tagged Questions

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### 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 ...
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### 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 $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
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### 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: ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Are faces of a compact, convex body “opposed” iff their extreme points are pairwise “opposed”?

Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W_1$, $W_2$ $\subset P$ are opposed if there ...
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### Altitudes of a triangle

The three altitudes of a triangle are concurrent -- this is true in all three constant curvature geometries (Euclidean, hyperbolic, spherical), but, as far as I know, the proofs are different in the ...
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### Uniform Embedding into Euclidean Space

Given a locally compact, separable, metric space $X$. When does $X$ uniformly embed into some Euclidean space? This means, when does there exist some integer $n$ and a closed subset ...
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### Least cardinality of a set of points in the plane

What is the least possible cardinality $K$, of a set S of points in the plane, such that there exists a point P in the plane and an open ball B centered at P, such that for all points X in B, not all ...
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### Discrete subgroups of isometry group of proper metric space

Let $X$ be a proper metric space and consider its isometry group $\mathrm{ISO}(X)$ endowed with the compact-open topology. Let $G$ be a subgroup of $\mathrm{ISO}(X)$. Consider the following ...
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### Boundaries of noncompact contractible manifolds

It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of ...
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### s-densities and conformal measures

I recently learnt about s-densities: http://en.wikipedia.org/wiki/Density_on_a_manifold#s-densities_on_a_vector_space For simplicity suppose that the vector space in this definition is \R. The prime ...
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### Local finiteness and coarse bounded geometry

I've just started learning these things and so probably my questions will be very easy. Please forgive me. A metric space $(X,d)$ is called locally finite if every bounded set is finite. A metric ...
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### Metric spheres in CAT(0) manifolds

Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric. Are sufficiently small metric spheres in $X$ homeomorphic to metric spheres in Euclidean space ...
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### Warped product neighborhoods of geodesic hypersurfaces

Is it true that any totally geodesic hypersurface in a nonpositively curved manifold has a tubular neighborhood such that the metric on the neighborhood is a warped product? At least, if the manifold ...
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### Is there a linear embedding of a simplical 3-complex in R^6?

I've heard that there always is an embedding in $R^7$ (can someone provide a reference for that?) and this number cannot be lowered in general. But I'm interested in a somewhat special case, namely: ...
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### Can a set of tetrahedra glued together by a common vertex be isometrically embedded in R^4?

A collection of triangles with a common vertex $A_1VA_2$, $A_2VA_3$, ... $A_NVA_1$ with specified side lengths can be isometrically embedded in $R^2$ provided the angles around $V$ add up to $2\pi$. ...
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### How do metrics behave under joining along a manifold embedded in the boundary?

How do metrics behave under joining along a manifold embedded in the boundary? This is, more-or-less, part of Problem 4.66 in Kirby's List: Problem 4.66 How do metrics (e.g. Riemannian, Lorentz, ...
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### Space-discriminating injective curve

Let $f\colon \mathbb R^1\to \mathbb R^3$ be a continuous and injective map. Is $\mathbb R^3\setminus f(\mathbb R^1)$ a path-connected space?
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### Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
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### How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?

Question 1. Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example ...
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### Explicit metrics

Every surface admits metrics of constant curvature, but there is usually a disconnect between these metrics, the shapes of ordinary objects, and typical mathematical drawings of surfaces. Can ...
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### G-spaces and manifolds

In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms: The space is metric The space is finitely compact, i.e., a ...
De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$ that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...