2
votes
0answers
104 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 ...
10
votes
1answer
198 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
1answer
215 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
0answers
403 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
1answer
125 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
0answers
384 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
1answer
196 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
0answers
122 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
1answer
208 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
1answer
239 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
2answers
242 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
2answers
323 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
1answer
259 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
2answers
606 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
2answers
369 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
1answer
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
3answers
295 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
0answers
253 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
1answer
569 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
1answer
691 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
1answer
596 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
2answers
291 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
2answers
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
0answers
564 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
2answers
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 ...