# Tagged Questions

**2**

votes

**0**answers

72 views

### Finitely generated groups non-embeddable into $L_1(0,1)$

I am interested in finitely generated groups which, endowed with their word metrics, do not admit bilipschitz embeddings into $L_1(0,1)$. I know two classes of such groups:
(1) Heisenberg group ...

**9**

votes

**1**answer

162 views

### Embeddings of finitely generated groups into uniformly convex Banach spaces

de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...

**5**

votes

**1**answer

210 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 ...

**1**

vote

**0**answers

368 views

### Bi invariant Riemannian metric on a Lie Group

I'm trying to find an example of a Lie group $G$ which admits a bi-invariant Riemannian metric, and which has a closed subgroup $H$ such that the manifold $G/H$ does not admit a $G$-invariant ...

**2**

votes

**1**answer

122 views

### Amalgmated free product of hyperbolic groups with one malnormal and one virtual factor is hyperbolic?

A possible formulation of the Bestvina-Feighn theorem is as follows (taken from here):
Combination Theorem (Bestvina & Feighn): If $H$ is a malnormal subgroup of hyperbolic groups $G_1, G_2$, ...

**7**

votes

**0**answers

377 views

### Uniquely geodesic groups

Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is ...

**2**

votes

**1**answer

193 views

### Divergence of geodesics in mapping class groups

I'm trying to learn some stuff about divergence of geodesics. Let $\gamma$ be a geodesic in a metric space $X$. The divergence of $\gamma$ is a function $f(r)$ for $r \ge 0$ such that $f(r)$ is the ...

**3**

votes

**0**answers

120 views

### Representing discrete groups in orthogonal groups

Suppose that we have a matrix $A$ of a quadratic form $Q_A$ of signature $(n,1)$ and a matrix $B$ of a quadratic form $Q_B$ which also has signature $(n,1)$. Let $O(Q_A)$ be the orthogonal group that ...

**5**

votes

**1**answer

205 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 ...

**11**

votes

**1**answer

233 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 ...

**1**

vote

**2**answers

233 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 ...

**0**

votes

**2**answers

317 views

### Why does the asymptotic cone fill the holes?

The first time I heard about the asymptotic cone, I ingenuously thought "Well... the asymptotic cone of $\mathbb Z^2$ minus the origin is $\mathbb R^2$ minus the origin". At that point somebody said ...

**1**

vote

**1**answer

255 views

### 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 ...

**12**

votes

**2**answers

593 views

### infinite dimensional CAT(0) groups

Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...

**8**

votes

**2**answers

363 views

### Quasi-isometry classes of elementary amenable groups

Is there any elementary argument showing that there exist uncountably many distinct quasi-isometry classes of elementary amenable groups? How about solvable groups?
For amenable groups it follows ...

**0**

votes

**1**answer

245 views

### Question about the proof of the fact that IR is not quasi-isomtric to IR^2 [closed]

Hello.
Yesterday we proofed, that $\mathbb{R}$ is not quasi-isometric to $\mathbb{R}^2$ (both endowed with the standard Euclidean metric).
Step 1.: $\mathbb{R}$ is q.i. $\mathbb{Z}$ and ...

**2**

votes

**3**answers

290 views

### 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 ...

**3**

votes

**0**answers

250 views

### Higher order Pansu derivative

Given a group $(G,*)$ there is no candidate for what can be understood as a derivative of a function $$f:G\rightarrow\mathbb{R}.$$ However, for the special case of Carnot groups there is the ...

**4**

votes

**1**answer

568 views

### Reformulation of amenability and growth rate of a group in terms of general metric spaces.

Updates: Changed a bit the definition to include infinite dimensional Banach spaces; Included questions 0 and subquestion. ...

**13**

votes

**1**answer

690 views

### Hearing the 17 planar symmetry groups

Though I'm sure it's really hard to work out for myself, does anyone know a reference for the spectra of the Laplacian on the 17 flat compact orbifolds that underlie the 17 planar symmetry groups.
...

**13**

votes

**1**answer

595 views

### Distance to an apartment of the affine building of GL(N)

Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...

**6**

votes

**2**answers

287 views

### Regularity of asymptotic cones

Are there any general conditions guaranteeing that the asymptotic cone of a group/graph is "regular" in some sense? E.g. for $\mathbb{Z}^d$ we get $\mathbb{R}^d$ as the asymptotic cone, which is even ...

**4**

votes

**2**answers

290 views

### 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 ...

**12**

votes

**0**answers

553 views

### Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive ...

**8**

votes

**2**answers

295 views

### Are there arbitrarily sparse “lattices” in negatively curved symmetric spaces?

Let $X$ be a negatively curved symmetric space. In other words, $X$ is one of the four examples: a hyperbolic space, a complex hyperbolic space, a quaternionic hyperbolic space or the hyperbolic ...