Knowledge of the behaviour of random walks on a fintiely generated group G carries a lot of information about intrinsic properties of G. The simplest and the most well-known demonstration of this principle is the Kesten's criterion of the amenability. However, even today, there are still many very natural questions surrounding this general phenomenon, remain widely open.
One such problem is the singularity conjecture. Due to H. Furstenberg and V. Kaimanovich, it is known that any random walk on a Fuchsian group $\Gamma$ with a finite first moment, and the support generating $\Gamma$, converges to its Gromov boundary. Moreover, we can identify the boudnary with $S^1$ in such a way, that the resulting hitting measure coincides with the harmonic measure on its Poisson boundary, which is the unique stationary measure with respect to the action of $\Gamma$. The singularity conjecture states that if, in addition, the random walk has finite support, the hitting measure is always singular with respect to the Lebesgue measure.
In this talk we will provide a survey of this conjecture, with the focus on the recent progress towards settling the conjecture for cocompact Fuchsian groups. Joint work with Giulio Tiozzo.