Questions tagged [coarse-geometry]

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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?
Anton Petrunin's user avatar
0 votes
0 answers
59 views

Can the sequence of complete graphs coarsely embed into Hilbert space?

Basically the title. If I have the metric space which is the disjoint union of the sequence of complete graphs, and the usual graph metric, has it been shown that the metric space can be coarsely ...
kreitz's user avatar
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4 votes
1 answer
173 views

Monoidal topology and coarse spaces

Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?
Cameron Zwarich's user avatar
1 vote
0 answers
73 views

Groups without "almost equivariant" coarse embeddings

Let $X$ be a set. We say that $\psi:X\times X\to[0,\infty)$ is a CND (conditionally negative definite) kernel if there is a Hilbert space $\mathcal{H}$ and a map $f:X\to\mathcal{H}$ such that \begin{...
I. Vergara's user avatar
2 votes
0 answers
127 views

Algebra of finite width matrices

$\DeclareMathOperator\FWM{FWM}\DeclareMathOperator\End{End}$For any ring $R$ there's an algebra of finite width matrices with entries in $R$. By finite width matrices I mean the ones that have only ...
Denis T's user avatar
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2 votes
0 answers
165 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 ...
Grisha Taroyan's user avatar
4 votes
0 answers
106 views

Alternative uniformities on topological groups

Are there any interesting alternative uniformities defined on topological groups besides the usual four (left, right, and their meet/join)? I am curious because in the (sort of) dual setting of coarse ...
Cameron Zwarich's user avatar
1 vote
0 answers
61 views

Consequences of having unbounded points in a bornology

For a set $X$ a bornology $\mathcal{B}$ is essentially an ideal in the power set $\mathcal{P}(X)$. Many sources including Wikipedia state additional property that $X = \bigcup \mathcal{B}$. Call it a ...
Nik Bren's user avatar
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1 vote
0 answers
91 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 ...
UserIn's user avatar
  • 103
4 votes
0 answers
150 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 ...
Math_Learner's user avatar
5 votes
1 answer
143 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 ...
jpmacmanus's user avatar
4 votes
0 answers
147 views

Ends of a metric space?

I'm looking for a definition of “ends” of a metric space that is well-defined even for non geodesic or locally finite metric spaces, invariant under quasi-isometries (or more generally coarse ...
user148575's user avatar
8 votes
0 answers
192 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 ...
Narutaka OZAWA's user avatar
13 votes
2 answers
746 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 ...
AGenevois's user avatar
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3 votes
0 answers
50 views

Dependence of Roe algebra and coarse index on the Riemannian metric

Let $(M,g)$ be a spin Riemannian manifold. The coarse index of the Dirac operator $D$ lies in the $K$-theory of the Roe algebra, which I will denote by $C^*(M,g)$ since its construction uses $g$. I ...
geometricK's user avatar
  • 1,851
7 votes
1 answer
267 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 $f:VX \to VY$ ($VX,\,VY$ ...
user148575's user avatar
5 votes
1 answer
169 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 $...
Christian Gorski's user avatar
0 votes
0 answers
231 views

The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation

Let $X$ be a coarse space, we define the following: $D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$ $f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
Hussain Rashed's user avatar
12 votes
0 answers
286 views

Topology is to semi-decidability, coarse structures are to what?

There is a folklore correspondence between topology as semi-decidability amongst computer scientists, which is explained in places like: The monograph Synthetic Topology: of Data Types and Classical ...
Siddharth Bhat's user avatar
5 votes
0 answers
108 views

Why are coarse maps required to be proper?

In the context of coarse spaces, a map between coarse spaces $f:X\to Y$ is called coarse if it is bornologous (it maps controlled sets to controlled sets), and proper, in the sense that preimages of ...
geodude's user avatar
  • 2,129
7 votes
0 answers
176 views

Coarsifying persistence modules

The context Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying: For all $t$ in $I$ but a closed discrete set of points $T$, there exists a ...
user148575's user avatar
3 votes
0 answers
125 views

Comparing the group convolution algebra with the equivariant Roe algebra

Let $G$ be a Lie group equipped with a left-invariant metric. Then $C_c(G)$ is a $*$-algebra of convolution operators on $L^2(G)$. Let $\mathbb{C}[|G|]^G$ denote the $*$-subalgebra of bounded ...
geometricK's user avatar
  • 1,851
2 votes
1 answer
176 views

Does the square root of a finite propagation operator have finite propagation?

Let $X$ be a non-compact manifold and let $C_0(X)$ act on $L^2(X)$ by pointwise multiplication. We say $T\in\mathcal{B}(L^2(X))$ has finite propagation if there exists an $r>0$ such that: for all ...
geometricK's user avatar
  • 1,851
0 votes
1 answer
79 views

Lower Estimate of A Lipschitz Map

Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function $\rho:(0,\infty)\...
ABIM's user avatar
  • 4,989
1 vote
1 answer
143 views

The product of two controlled operators is also a controlled operator

The following picture is lemma 4.23 in Lectures on Coarse Geometry by John Roe: I guess the $E_i$ in the centered formula is $X_i$. Does Roe mean that $X_j\cap \mathrm{Supp}(u)=\emptyset $ implies $\...
C. Ding's user avatar
  • 135
1 vote
2 answers
437 views

How to choose a continuous function which vanishes **only** on the closed set

We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9: Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a ...
C. Ding's user avatar
  • 135
5 votes
1 answer
310 views

Reference request: Higson compactification

It seems that the idea of the Higson compactification first arose in the context of non-compact manifolds in a 1992 preprint of Higson called "The relative $K$-homology of Baum and Douglas". It seems ...
geometricK's user avatar
  • 1,851
6 votes
1 answer
178 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 ...
Jake Herndon's user avatar
4 votes
1 answer
150 views

Coarsely trivial Borel cross section for $G\to G/N$

Let $G$ be a locally compact group, and let $N$ be a closed, normal subgroup, and let $\pi\colon G\to G/N$ be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a ...
Hannes Thiel's user avatar
  • 3,305
3 votes
2 answers
285 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.
user44172's user avatar
  • 541
11 votes
1 answer
288 views

Duality between large and small scale structures

A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a ...
Jamie Walton's user avatar
13 votes
2 answers
262 views

Is $\mathbb{H}^n$ quasi-isometric to a leaf of a codimension 1 foliation of a compact manifold?

If we extend the action of $\pi_1(\Sigma_g), g\geq 2,$ from $\mathbb{H}^2$ to its boundary $\partial_{\infty}\mathbb{H}^2=S^1$, the surface bundle corresponding to this action of $\pi_1(\Sigma_g)$ on $...
Robert Schmidt's user avatar
8 votes
0 answers
181 views

Can two random graphs be metrically embedded into one another?

Let $X, Y$ be two random graphs on $n$ vertices (say, in $G(n, p)$ model for some $p$). Can anything (expectation, value with high probabiity, ...) be said about $D(X, Y)$, where $D$ is the minimal ...
Marcin Kotowski's user avatar