A recurring question in the theory of random walks on groups of isometries of hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift, also known as the "fundamental" inequality.

For certain finitely-supported random walks on cocompact Fuchsian groups, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1.

Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.

Joint with P. Kosenko.