## Quasi-isometries and stability of quasi-geodesics

Continuation of expository series on Gromov hyperbolicity. Recall that a map ${f\colon X\rightarrow Y}$ is a quasi-isometry if there are constants ${L,M}$ such that ${L^{-1}|x_1x_2|-M\le |f(x_1)f(x_2)|\le L|x_1x_2|+M}$ for all ${x_1,x_2\in X}$. This is a coarse version of the bi-Lipschitz condition. Surprisingly, Gromov hyperbolicity is preserved under quasi-isometries of geodesic spaces. The surprising part is that the multiplicative constant ${L}$ does not kill the additive constant ${\delta}$.

Theorem. Suppose ${X}$ and ${Y}$ are geodesic metric spaces, and ${Y}$ is Gromov hyperbolic. If there exists a quasi-isometry ${f\colon X\rightarrow Y}$, then ${X}$ is also Gromov hyperbolic.

Proof goes like this. Assuming that ${X}$ contains a fat geodesic triangle ${a,b,c}$, we consider the geodesic triangle in ${Y}$ with vertices ${f(a),f(b),f(c)}$, and want to prove that it is also fat. Since ${f}$ is a quasi-isometry, it follows that the images of geodesics ${ab}$, ${bc}$ and ${ac}$ form a roughly-triangular shape which has the fatness property: there is a point on one of the sides that is far away from the other two sides. The problem reduces to showing that this roughly-triangular shape lies within a certain distance ${R}$ (independent of ${a,b,c}$) from the actual geodesic triangle with vertices ${f(a),f(b),f(c)}$. This is known as stability of quasi-geodesics. A quasi-geodesic is a quasi-isometric image of a line segment, similar to how a geodesic is a (locally) isometric image of a segment.

By the way, quasi-geodesic stability fails in ${\mathbb R^2}$. We can connect the points ${(-n,0)}$ and ${(n,0)}$ by the quasi-geodesic ${y=n-|x|}$, which is at distance ${n}$ from the true geodesic between these points.

I’ll prove a more specialized and weaker statement, which however contains the essence of the full result. Namely, let $\mathbb H^2$ denote the hyperbolic plane and assume that ${f\colon [a,b]\rightarrow \mathbb H^2}$ is bi-Lipschitz: ${L^{-1}|x_1-x_2|\le |f(x_1)f(x_2)|\le L|x_1-x_2|}$ for all ${x_1,x_2\in [a,b]}$. The claim is that the image of ${f}$ lies in the ${R}$-neighborhood of the geodesic through ${f(a)}$ and ${f(b)}$, where ${R}$ depends only on ${L}$.

There are three standard models of hyperbolic plane: disk, halfplane and infinite strip. I’ll use the last one, because it’s the only model in which a geodesic is represented by Euclidean line. Specifically, ${\mathbb H^2}$ is identified with the infinite strip ${\{x+iy\in\mathbb C\colon |y|<\pi/2\}}$ equipped with the metric ${|dz|/\cos y}$. (To see where the metric comes from, apply $z\mapsto i e^{iz}$ to map the strip onto upper halfplane and pull back the hyperbolic metric $|dw|/\mathrm{Im}\,w$.)

The hyperbolic and Euclidean metrics coincide on the real line, which is where we place ${f(a)}$ and ${f(b)}$ with the help of some hyperbolic isometry. Let ${\Gamma=f[a,b]}$ be our quasi-geodesic. Being a bi-Lipschitz image of a line segment, ${\Gamma}$ satisfies the chord-arc condition: the length of any subarc of ${\Gamma}$ does not exceed ${L^2}$ times the distance between its endpoints. Pick ${y_0\in (0,\pi/2)}$ such that ${1/\cos y_0=2L^2}$. Let ${D}$ be the hyperbolic distance between the lines ${y=y_0}$ and ${y=0}$. This distance could be calculated as ${\int_0^{y_0}\sec y\,dy}$, but I’d rather keep this integral as an exquisite Calculus II torture device.

The problem facing us is that quasigeodesic may be $L^2$ times longer than the distance between its endpoints, which seems to allows it to wander far off the straight path. However, it turns out there is a uniform bound on the length of any subarc ${\Gamma'}$ of ${\Gamma}$ that lies within the substrip ${\{y \ge y_0\}}$. We lose no generality in assuming that the endpoints of $\Gamma'$ are on the line $y=y_0$; they will be denoted ${x_j+iy_0}$, ${j=1,2}$. The key point is that connecting these two points within $\{y\ge y_0\}$ is rather inefficient, and such inefficiency is controlled by the chord-arc property.

The hyperbolic distance between ${x_j+iy_0}$ is at most $|x_1-x_2|+2D$, because we can go from ${x_1+iy_0}$ to ${x_1}$ (distance ${D}$), then from ${x_1}$ to ${x_2}$ (distance ${|x_1-x_2|}$), and finally from ${x_2}$ to ${x_2+iy_0}$ (distance ${D}$). On the other hand, the length of ${\Gamma'}$ is at least ${|x_1-x_2|/\cos y_0 = 2L^2|x_1-x_2|}$ because the density of hyperbolic metric is at least $1/\cos y_0$ where $\Gamma'$ lives. The chord-arc property yields ${2L^2 |x_1-x_2| \le L^2 (|x_1-x_2|+2D)}$, which simplifies to ${|x_1-x_2| \le 2D}$. Hence, the distance between the endpoints of ${\Gamma'}$ is at most ${4D}$, and another application of the chord-arc property bounds the length of ${\Gamma'}$ by ${4DL^2}$.
In conclusion, the claimed stability result holds with ${R= D+2DL^2}$.