Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $SL(2, \mathbb R)$ to $\mathbb H$ which is defined by $$A \cdot z = \frac{az+b}{cz+d}, \ \ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL(2, \mathbb R), \ z \in \mathbb H. $$ Specifically, $$ d(A \cdot z, A \cdot w) = d(z,w), \ \ z, w \in \mathbb H, A \in SL(2, \mathbb R).$$ Then, without assuming that $(\mathbb H,d)$ is geodesic, can we say that $d$ is Gromov-hyperbolic? Our original motivation comes from the square roots of some $f$-divergences between univariate Cauchy distributions which are non-geodesic metrics on $\mathbb H$.

$\DeclareMathOperator\SL{SL}$Let $d$ be a metric on the upper-half plane $\mathbb H = \{(x,y) : y > 0\}$ which is invariant with respect to the action of $\SL(2, \mathbb R)$ to $\mathbb H$ which is defined by $$A \cdot z = \frac{az+b}{cz+d}, \ \ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \SL(2, \mathbb R), \ z \in \mathbb H. $$ Specifically, $$ d(A \cdot z, A \cdot w) = d(z,w), \ \ z, w \in \mathbb H, A \in \SL(2, \mathbb R).$$ Then, without assuming that $(\mathbb H,d)$ is geodesic, can we say that $d$ is Gromov-hyperbolic? Our original motivation comes from the square roots of some $f$-divergences between univariate Cauchy distributions which are non-geodesic metrics on $\mathbb H$.