I was going through the books Géométrie et théorie des groupes by Michel Coornaert, Thomas Delzant, Athanase Papadopoulos and Metric Spaces of Non-Positive Curvature by Martin R. Bridson, André Haefliger. These books give different definitions of quasi-geodesics. In the former,
Let$\ X$ be a metric space and$\ f: [a,b] \to X$ a continuous path, with$\ - \infty \leq a \leq b \leq \infty.$ We say that$\ f$ is a$\ (\lambda,k)$-quasi-geodesic (where$\ \lambda \geq 1$ and$\ k \geq 0$) if for every sub-interval$\ [a',b']$ of$\ [a,b],$ we have: \begin{equation} l(f([a',b'])) \leq \lambda |f(a') - f(b')| + k. \end{equation} We say that$\ f$ is a global quasi-geodesic, or simply a quasi-geodesic, if there exists a couple$\ (\lambda,k)$ such that$\ f$ is a$\ (\lambda,k)$-quasi-geodesic. We finally say that$\ f$ is a$\ (\lambda,k,L)$-local-quasi-geodesic(where$\ \lambda$ and$\ k$ are as above, and$\ L>0$) if for every sub-interval$\ [a',b']$ of$\ [a,b]$ such that$\ l(f([a',b'])) \leq L,$ we have: \begin{equation} l(f([a',b'])) \leq \lambda |f(a') - f(b')| + k. \end{equation}
I am also attaching the image of the definition below, from the book while in the latter, the definition is,
A$\ (\lambda, \epsilon)$-quasi-geodesic in a metric space$\ X$ is a$\ (\lambda,\epsilon)$-quasi-isometric embedding$\ c : I \to X$, where$\ I$ is an interval of the real line (bounded or unbounded) or else the intersection of$\ \mathbb{Z}$ with such an interval. More explicitly, \begin{equation*} \frac{1}{\lambda} d(t,t') - \epsilon \leq d(c(t),c(t')) \leq \lambda d(t,t') + \epsilon \end{equation*} for all$\ t, t' \in I$. If$\ I = [a, b]$ then$\ c(a)$ and$\ c(b)$ are called the endpoints of$\ c$.
Are these definitions equivalent? If so could you please show it?
