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TL,DR: Why do we have $$d(\overline{\kappa} \circ \kappa, Id) \leq 32 \delta?$$


I am reading Chapter 11 from the book "Geometric group theory" by Cornelia Druţu and Michael Kapovich (freely available here: https://www.math.ucdavis.edu/%7Ekapovich/EPR/ggt.pdf) and I have trouble understanding the end of section 11.8.

The setting is the following, we work in a $\delta$-hyperbolic geodesic space $X$ (in the sense of Rips), and we want to study geodesic triangles in $X$. For this, the authors define a collapsing map $\kappa$ that allows us to compare these triangles with geodesic triangles in a real tree, i.e. tripods (like one would do to study $CAT(\kappa)$ spaces).

The authors define $\kappa$ as follows: "Now, given a geodesic triangle $T = T (x_1 , x_2 , x_3 )$ with side-lengths $a_i$, $i = 1, 2, 3$ in a metric space $X$, there exists a unique (possibly up to postcomposition with an isometry $\tilde{T}\to \tilde{T}$ ) map $\kappa$ to the comparison tripod $\tilde{T}$, $$\kappa : T \to \tilde{T} = \mathcal{T}_{a_1 ,a_2 ,a_3} ,$$ which restricts to an isometry on every edge of T : The map $\kappa$ sends the vertices $x_i$ of $T$ to the leaves $\tilde{x_i}$ of the tripod $\tilde{T}$. The map $\kappa$ is called the collapsing map for $T$ . We say that the points $x, y \in T$ are dual to each other if $\kappa(x) = \kappa(y)$."

Question 1: From the book, p.385: "Let $p_3 \in \gamma_{12} = x_1x_2$ be a point closest to $x_3$. Taking $R = 2\delta$ and combining Lemma 11.22 with Lemma 11.52, we obtain:
Corollary 11.62: $d(p_3, x_{12}) \leq 2(2\delta + 2\delta) = 6\delta$"

  • First of all, it should be $8 \delta$ and not $6 \delta$.
  • Moreover, I can't prove $6 \delta$ or $8 \delta$. The best I can do is $16 \delta$. Can anyone prove the previous corollary?

Still from the book, p.385: "We now can define a continuous coarse inverse $\overline{\kappa}$ to $\kappa$ as follows : We map the geodesic segment $\tilde{x_1}\tilde{x_2} \subset \tilde{T}$ isometrically to a geodesic $x_1x_2$. We send $o\tilde{x_3}$ onto a geodesic $x_{12}x_3$ by an affine map. Since $$d(x_{12},x_{32}) \leq 6\delta$$ and $$d(x_3, x_{32}) = d(\tilde{x_3},o),$$ we conclude that the map $\overline{\kappa}$ is $(1, 6\delta)-Lipschitz.
Exercise 11.63:$$d(\overline{\kappa} \circ \kappa, Id) \leq 32 \delta.$$

Question 2:

  1. Why do they say "We send $o\tilde{x_3}$ onto a geodesic $x_{12}x_3$ by an affine map.", and what do they mean by "affine map"?
  2. What about the other sides of the original triangle? Do we get a map like the first picture below (which is how I understand the construction), or like the second picture (which would make more sense)?
  3. Again, I can't solve the exercise. I can prove that there exists some $N$ such that this is smaller than $N \delta$, but I can't prove that $N = 32$. Any idea?

Let me know if I need to provide more information, and thanks in advance for any help.

<span class=$\kappa$ and $\overline{\kappa}$"> <span class=$\kappa$ and $\overline{\kappa}$">

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Regarding your questions about constant multiples of the hyperbolicity constant $\delta$, what's usually important in these proofs is existence of some $C$ such that a certain inequality of the form $\ldots \le C\delta$ is true. It's an interesting game to try to get a better and better value of $C$, and for certain situations the optimal value of $C$ might be important or at least useful. But that kind of situation is rarer than you think. I would say that if the best you can do is $16\delta$ then that's just fine.

Regarding your question about affine maps, if one is given two geodesics $\gamma : [a,b] \to X$ and $\delta : [c,d] \to Y$ in two spaces, an affine map from the subset $\gamma[a,b] \subset X$ to the subset $\delta[c,d] \subset Y$ is a homeomorphism $F : \gamma[a,b] \to \delta[c,d]$ which can be expressed in the form $F(\delta(t)) = \gamma(f(t))$ where $f : [a,b] \to [c,d]$ is affine in the ordinary sense, meaning $f(t)=\alpha t + \beta$ for appropriately chosen constant $\alpha,\beta$. An equivalent way to put this is that the map $F$ stretches distance by constant factor: $F(\delta(s),\delta(t)) = K |s-t|$ for some constant $K>0$ independent of $s,t$.

Regarding the other two sides, it doesn't really matter. All that matters is that you get a map which is a coarse inverse having some values of the constants; obtaining the optimal values of the constants is rarely important.

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