The moduli space of quasimorphisms on subshifts and its pressure geometry

We introduce a metric geometry on the space of positive, linearly growing quasimorphisms on a mixing subshift of finite type. Two such quasimorphisms are identified when their homogenizations agree up to multiplicative scaling. The resulting moduli space carries a natural symmetrised Thurston-type metric, defined from the extremal ratios of their periodic orbit lengths. We prove that this is a genuine metric and determine several global features of the resulting space: it is unbounded, separable, incomplete, and has geodesics that are canonically generated by thermodynamic pressure functions. We then describe the endpoints of these geodesics and develop a boundary theory for our moduli space. We apply this theory to metric structures on right-angled Artin and Coxeter groups. In particular we construct isometric embeddings of geometric moduli spaces into our quasimorphism moduli spaces. As a consequence, Manhattan curves for metrics on these groups are $C^1$, with equality cases characterised by rough similarity of the underlying metrics. This provides insights into Manhattan curves in settings where neither classical thermodynamic formalism for Hölder potentials nor Patterson-Sullivan theory apply. The same construction recovers the corresponding results for hyperbolic groups and places these examples within a common dynamical-geometric framework.