Questions tagged [coarse-geometry]
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33
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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?
4
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1
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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?
0
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59
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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 ...
1
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0
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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{...
2
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0
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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 ...
2
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0
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166
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Characterization of growth in terms of coarse algebraic topology
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\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 ...
4
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0
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106
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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 ...
1
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0
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61
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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 ...
1
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0
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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 ...
4
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0
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152
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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 ...
5
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1
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145
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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 ...
4
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0
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150
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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 ...
8
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194
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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 ...
13
votes
2
answers
752
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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 ...
3
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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 ...
7
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1
answer
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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$ ...
5
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1
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171
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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 $...
7
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176
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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 ...
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0
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233
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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 ...
12
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0
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286
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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 ...
5
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0
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108
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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 ...
3
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0
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125
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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 ...
2
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1
answer
177
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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 ...
0
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1
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79
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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)\...
11
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1
answer
288
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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 ...
1
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1
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143
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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 $\...
1
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2
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438
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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 ...
5
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1
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310
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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 ...
6
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1
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178
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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 ...
4
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1
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151
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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 ...
3
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2
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285
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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.
13
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2
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262
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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 $...
8
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0
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181
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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 ...