3

Let $(M_1,d_1)$ and $(M_2,d_2)$ be two metric spaces. We say that these two spaces are quasi-isometric if there exists a map $f$ between them which satisfies the following :

$$ \forall x,y\in M_{1} :{\frac{1}{A}}d_{1}(x,y)-B\leq d_{2}(f(x),f(y))\leq A d_{1}(x,y)+ B,$$

and $$ \exists C \geq 0, \forall z\in M_{2}:\exists x\in M_{1}:d_{2}(z,f(x))\leq C.$$

I want to understand this definition intuitively: according to wikipedia a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details, could anyone elaborate further on this or (and) give a more intuitive explanation of this definition please?

sxccalmat1100
  • 440
  • 2
  • 10
Asma
  • 371
  • 3
    With this definition, the line $\Bbb R$ and the discrete set of aligned points $\Bbb Z$ are quasi-isometric. The intuition is: draw $\Bbb Z$ on a blackboard, then go away from it, and look at it. You will see a line, at least coarsly – Didier Feb 03 '24 at 10:05

1 Answers1

2

As Didier commented, if you draw $\Bbb Z$ on a blackboard, you will have some dots placed horizontally. As you move away from the blackboard, the dots will appear closer, and finally, when you are at a sufficiently large distance from the blackboard, you will effectively see a continuous line, i.e. $\Bbb R$. The space $\Bbb Z$ by itself is not very interesting, it is a discrete space. When we look at it from far away, it has some interesting properties. Note that $\Bbb N$ is not quasi-isometric to $\Bbb Z$ (Why?). The essential principle of large-scale geometry is that you look at some object, then go "far away" and have a look at it again. What interesting properties does this object have now? All metric spaces that look similar when you are "far away" from them can thought to be quasi-isometric. Quasi-isometric metric spaces have the same "geometric properties". Later on, you will study notions like hyperbolicity, which are preserved under quasi-isometries. These "geometric properties" are termed as quasi-isometric invariants. Quasi-isometries are weakened bilipschitz maps. They can be thought of as discontinuous maps over whose discontinuities you have some amount of "control".

sxccalmat1100
  • 440
  • 2
  • 10
  • thank you very for your answer! but Why (or how) does this idea translated mathematically into the two conditions in the question ? – Asma Feb 03 '24 at 11:50
  • A map satisfying the first condition is said to be a quasi-isometric embedding. If a map as defined by you exists between $M_1$ and $M_2$, it is equivalent to saying that there exists a quasi-isometric embedding $g$ from $M_1$ to $M_2$ and another q-i embedding $h$ in the opposite direction, and additionally, the compositions of the q-i embeddings are at a finite distance from $Id_{M_1}$ and $Id_{M_2}$. Effectively, what you have is that there may not exist a bijection between the two spaces, but something "close" to a bijection, and the discontinuities of this map is under "control". – sxccalmat1100 Feb 03 '24 at 13:37
  • Crudely saying, at a distance "far away", you have something akin to a continuous bijection, and its "inverse", is also "continuous" when seen from "afar". Thus, effectively, they are "homeomorphic". – sxccalmat1100 Feb 03 '24 at 13:42
  • .@sxccalmat, thanks again! sorry that my comment wasn't very clear, what actually my question was is why does this idea of ''far way" translated mathematically by multpling the distance on the right by $A$ and adding $B$, and by multpling it on the left by $\frac{1}{A}$ and adding $-B$? – Asma Feb 03 '24 at 15:45
  • 1
    Suppose $B$ is $0$. Then it is a continuous map (in fact bilipschitz). Mind you, the first condition is an inequality on the real line. If $B$ is non zero, the function may not be continuous. Now, visualise the real line. As you move far away, will that $B$ make any difference to the continuity? For more information, you should check chapter 5 of Clara Loeh's book on Geometric Group Theory. I think that will answer your questions. – sxccalmat1100 Feb 03 '24 at 16:32