# The collapsing map and its coarse inverse are $32 \delta$-coarse Lipschitz to each other

## 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.

## 1 Answer

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.