Short map between hyperbolic triangles

Question

Given two (Euclidean or hyperbolic) triangles $ T = ABC $ and $ T' = A'B'C' $, the natural map is the one that sends $ A' \mapsto A $, $ B' \mapsto B $, $ C' \mapsto C $ and maps affinely each side of $T'$ onto the corresponding side of $T$. We say that the triangle $ T' $ dominates the triangle $ T $ if the natural map is a short map (Lipschitz with constant $1$) with respect to the distance in the (Euclidean or hyperbolic) plane.

My question is, given a triangle $T$ with side lengths $(a, b,c)$, is it true that the triangle $T'$ with side lengths $(a+\epsilon, b+\epsilon, c+\epsilon)$ dominates $T$ for all $\epsilon>0$ small enough?

I can prove this statement for Euclidean triangles by some calculations involving the law of cosines, but I couldn't manage to do the same in the hyperbolic plane.

Answer 1

No, this is not true. Here is an indirect argument (if I made no mistake).

If the statement would be true for $\epsilon > 0$ small enough, then it would be true for all $\epsilon > 0$ (since the maps commute, the set of "good" epsilons for a given $(a,b,c)$ is open; this set is also closed because the Lipschitz condition is closed). We will construct two triangles $(a,b,c)$ and $(a+M, b+M, c+M)$ for which the latter does not dominate the former.

Take $a=b=1$ and $c=2$ (or slightly smaller if you need). Take points $A_1$ and $B_1$ on the sides $BC$ and $AC$ at distance $\frac13$ from $C$. The distance between $A_1$ and $B_1$ is $\frac23$. Now consider a triangle $(x, x, x+1)$ for a very large $x$. I claim that the distance between the corresponding points $A'_1$ and $B'_1$ tends to $0$ as $x$ tends to $\infty$.

By the sine law in a right triangle we have $$\frac{\sinh \frac{A'_1B'_1}{2}}{\sinh\frac{x}{3}} = \sin\frac{\gamma'}{2} = \frac{\sinh\frac{x+1}2}{\sinh x}$$ which implies that $\sinh\frac{A'_1B'_1}2$ goes down as $e^{-\frac{x}6}$.

This means that somewhere inbetween there is an isosceles triangle for which the map is not $1$-Lipschitz even for small $\epsilon$...

Answer 2

We can answer the question positively if we adapt the definition of domination to the hyperbolic setting. Given two hyperbolic triangles $ T = ABC $ and $ T' = A'B'C' $, the natural map is the one that sends $ A' \mapsto A $, $ B' \mapsto B $, $ C' \mapsto C $ and maps each side of $T'$ onto the corresponding side of $T$ with the following parametrization:

A geodesic $[x,y]$ of length $l$ is in bijection with the interval $[0,1]$ via the map that sends $t$ to the point in $[x,y]$ at distance $\sinh^{-1} ( t \sinh (l))$ from $x$; now maps each side of $T'$ to the corresponding side of $T$ with the mapping which sends a point to the point with the same parameter in $[0,1]$.

With this definition of domination it can be check directly using the hyperbolic law of cosines that for small epsilon, the triangle with sides $(a+\epsilon, b+\epsilon, c+\epsilon)$ dominates the triangle with sides $(a,b,c)$.