There is a bijection $f$ between $PSL(2,\mathbb{R})$ and $T^1\mathbb{H}$, which sends a matrix $M$ to a vector with base point $\frac{M_{11}i+M_{12}}{M_{21}i+M_{22}}$. In particular, the identity matrix $I$ is sent to a vector with base point which is the imaginary unit $i$, and points upwards.
Let $\pi$ be the projectuon from $T^1\mathbb{H}$ to $\mathbb{H}$ sending a vector to its basepoint.
I was wondering if given the hyperbolic distance between $\pi f(M)$ and $i$, one could deduce the distance between $M$ and $I$. I am most interested in being able to bound from above the entries of $M$.
First, it makes sense that if $\pi f(M)$ is in some neighborhood of $i$ then $M$ is in some neighborhood of $I$ and in particular $M_{ij}$ is bounded.
Quantitatively, my intuition says that $|M_{ij}-\delta_{ij}|<const\cdot\exp({d(\pi f(M),i)})$ should hold, giving a nice correspondence between the maximum norm on $PSL(2,\mathbb{R})$ and the hyperbolic metric (namely $\|M-I\|_{max}<C\cdot\exp(d(\pi f(M),\pi f(I)))$. Is it true?
Evidence for that is the easy to calculate case of a diagonal $M$ with $x,\frac{1}{x}$ on the diagonal (for $x>1$), which correspondes to a vector with basepoint $x^2i$ for which $d(x^2i,i)=log(x^2)$. Yet, calculation with more complicated matrices (using the Iwasawa decomposition for instance) has not fully worked for me. I tried using the hyperbolic metric explicitly, obtaining some equality that does not seem to hold if the entries of $M$ are indeed too large, but it is "messy" and I wonder if there is some easier argument (or if my way can be finished without too many technical difficulties).