# Questions tagged [coarse-geometry]

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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{...
115 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 ...
157 views

### Characterization of growth in terms of coarse algebraic topology

$$\newcommand{\mc}{\mathcal{#1}} \newcommand{\mbb}{\mathbb{#1}} \newcommand{\opn}{\operatorname{#1}} \DeclareMathOperator\cap{cap} \def\sse{\subseteq}$$ Coarse spaces Let $X$ be a coarse ...
103 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 ...
1 vote
54 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 ...
1 vote
89 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 ...
127 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 ...
129 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 ...
134 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 ...
177 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 ...
681 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 ...
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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 ...
253 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 $f:VX \to VY$ ($VX,\,VY$ ...
149 views

1 vote
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 $\... 1 vote 2 answers 385 views ### How to choose a continuous function which vanishes **only** on the closed set We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9: Suppose$X$is a paracompact and locally compact Hausdorff space,$\bar{X}$is a ... 5 votes 1 answer 293 views ### Reference request: Higson compactification It seems that the idea of the Higson compactification first arose in the context of non-compact manifolds in a 1992 preprint of Higson called "The relative$K$-homology of Baum and Douglas". It seems ... 6 votes 1 answer 175 views ### The growth of a subset of a group Let$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$then it ... 4 votes 1 answer 149 views ### Coarsely trivial Borel cross section for$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$be the quotient homomorphism. It is known that there exists a Borel cross section, i.e., a ... 3 votes 2 answers 277 views ### F.g group with infinite ends not Q.I to a free group Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group? Thanks in advance. 11 votes 1 answer 270 views ### Duality between large and small scale structures A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure$\mathcal{C}$(defined by a ... 13 votes 2 answers 258 views ### Is$\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)$on$...
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 ...