Given a finitely presented group $(G, S)$, a left-invariant distance $d$ on $G$ and a nice enough random walk $(X_n)$ on $G$, we can associate three invariants: the Avez entropy $h$, the drift $l$, and the logarithmic volume $v$. The fundamental inequality states that $h \leq l \cdot v$ whenever all three invariants are well-defined.
However, we don't always end up having an equality $h = l \cdot v$, and classifying all triples $(G, d, (X_n))$ for which we have the equality seems extremely difficult. Moreover, even if we restrict ourselves to the case when G is a Fuchsian group generated by the side-pairing transformations relative to its fundamental polygon and set $d$ to be the word metric or the hyperbolic metric induced by the action on $H^2$, the problem still remians quite tricky.
In this talk we will discuss the recent progress made in the case of the hyperbolic metric on cocompact Fuchisan groups. We will give all necessary definitions, explain the relation between the fundamental inequality and Patterson-Sullivan theory, and, if time permits, we will briefly talk about the ideas used in the proofs themselves. Joint work with G. Tiozzo.