All Questions
Tagged with geometric-group-theory coarse-geometry
12 questions
23
votes
1
answer
1k
views
Universal graph
A connected (and infinite) graph $U$ will be called $n$-universal if any connected graph with degree $\leqslant n$ admits an embedding in $U$.
Is there a 3-universal graph with bounded degree?
- 45k
13
votes
2
answers
807
views
Prehistory of Gromov-hyperbolic spaces/groups
When speaking about hyperbolic groups/spaces, one usually refers to Gromov's monograph Hyperbolic groups for their introduction. However, coarse notions of hyperbolicity can be found in some of his ...
- 8,401
8
votes
0
answers
228
views
Coarse quotient maps
Interesting connections and analogies have been observed between
non-linear geometry of Banach spaces and coarse geometry.
In the former subject, people have investigated the notion of
uniform (or ...
- 10.1k
7
votes
1
answer
283
views
Are two quasi-isometric, isomorphic on large enough balls, transitive graphs isomorphic?
Take two transitive graphs $X,Y$ (potentially directed and edge-labelled, e.g. Cayley graphs).
Assume $X,Y$ are quasi-isometric with constant $K$, i.e. there exists a function $f:VX \to VY$ ($VX,\,VY$ ...
- 887
6
votes
1
answer
179
views
The growth of a subset of a group
Let $S$ be a symmetric subset of a group $G$ containing the identity, and let $S^n$ be the set of all products of $n$ elements of $S$. If $S^3\subset gS$ for some translate $gS$ of $S$ then it ...
- 63
5
votes
1
answer
171
views
Given a quasi-convex subgroup $H$ of hyperbolic $G$, can we decide if two elements $x,y \in G$ lie in the same double coset of $H$?
I've come across the following question in my research, which seems elusive but is almost surely decidable.
Let $H$ be a quasi-convex subgroup of the hyperbolic group $G$. Given $x, y \in G$, we wish ...
- 629
5
votes
1
answer
200
views
Example of an invariant metric on a nilpotent group which is not asymptotically geodesic
Let $X$ be a metric space. We say that $X$ is asymptotically geodesic if for all $\epsilon > 0$, there exists $R > 0$ such that, for all $x,y \in X$, there exists some finite sequence of points $...
- 305
4
votes
0
answers
83
views
Additive characters from coarse quotient maps
Let's consider a (finitely generated) group $\Gamma$ and a
coarse quotient map
$q\colon\Gamma\to\mathbb{R}$.
I'm interested in the 1-cocycle
$\sigma\colon\Gamma\to\ell_\infty\Gamma$,
defined by $\...
- 10.1k
4
votes
0
answers
186
views
Ends of a negatively curved Riemannian manifold
Let $M$ be a complete Riemannian manifold. Let us use the standard definition of "end", for example, as in this article. If $M$ has non-negative Ricci curvature, it is well-known that it has ...
- 41
3
votes
2
answers
287
views
F.g group with infinite ends not Q.I to a free group
Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group?
Thanks in advance.
- 541
2
votes
0
answers
171
views
Characterization of growth in terms of coarse algebraic topology
$$
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mbb}[1]{\mathbb{#1}}
\newcommand{\opn}[1]{\operatorname{#1}}
\DeclareMathOperator\cap{cap}
\def\sse{\subseteq}
$$
Coarse spaces
Let $X$ be a coarse ...
- 1,374
1
vote
0
answers
98
views
Question about coarse fixed point property in large-scale geometry
I read the article of Steven Hair "A degree-theoretic proof of a coarse fixed point principle". I have the following question.
I start with some main definitions from this article. A coarse ...
- 103