Questions tagged [coarse-geometry]

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12
votes
2answers
487 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 ...
2
votes
0answers
27 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 ...
7
votes
1answer
207 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 $f:VX \to VY$ ($VX,\,VY$ ...
6
votes
1answer
92 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 $...
0
votes
0answers
142 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 ...
11
votes
0answers
190 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 ...
5
votes
0answers
77 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 ...
6
votes
0answers
158 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 ...
3
votes
0answers
81 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 ...
2
votes
1answer
144 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 ...
0
votes
1answer
65 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)\...
1
vote
1answer
134 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 $\...
1
vote
2answers
289 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
1answer
236 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
1answer
160 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
1answer
137 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
2answers
271 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
1answer
233 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
2answers
247 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 $...
8
votes
0answers
166 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 ...