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“Are all gaps there?” - a question originally posed by Mark Kac for the Almost-Mathieu operator and later termed the Dry Ten Martini Problem by Simon - asks whether all gap labels predicted by the Gap Labelling Theorem are attained by the integrated density of states. For the Almost-Mathieu operator, this problem has been resolved in almost all regimes, and for Sturmian Hamiltonians it was recently answered affirmatively, with all allowed gaps shown to be open. In this talk, we introduce a geometric analogue of this problem for operators on discrete and metric graphs generated by Sturmian subshifts, where the aperiodicity is encoded in the underlying graph structure rather than in a potential. We study ergodic Jacobi operators on such graphs and compute their gap labels explicitly. In contrast to the classical Sturmian Hamiltonians, we show that not all allowed gaps are open. The missing gaps arise from a geometric mechanism rather than a dynamical one: the local structure of the graph produces flat bands, leading to unattained gap labels. Based on joint work with Ram Band.

In this talk, I will discuss a joint work with Samuel J. Taylor, where we consider embedded non-fiber surfaces in a closed hyperbolic 3-manifold transverse to a pseudo-Anosov flow. Utilizing foundational work on Kleinian surface groups, we connect the hyperbolic geometry of the surface to the dynamics of the flow via the curve graph. More specifically, the flow gives a pair of special multicurves on the surface, and we show that they are also geometrically distinguished, enabling us to extract geometric information, such as the position of short curves and the volume, from the relation of the pair of multicurves in the curve graph. Our result also connects to recent study of the geometry of endperiodic mapping tori.