  We investigate the dynamic properties of elastic lattices defined by
tessellations of a curved hyperbolic space. The lattices are obtained by
projecting nodes of a regular hyperbolic tessellation onto a flat disk and then
connecting those sites with simple linkages. Numerical and experimental
investigations illustrate how their vibrational spectral properties are
characterized by a high density of modes that are localized at the domain
boundaries. Such properties govern the propagation of waves induced by
broadband inputs. This suggests the potential for applications seeking the
protection of bulk media from boundary-incident perturbations. We uncover the
boundary-dominated nature of an exemplary hyperbolic lattice through the
evaluation and analysis of its integrated density of states and vibrational
spectrum. The dynamics of the lattice are also contextualized by comparing them
with those of continuous disks characterized by Euclidean and hyperbolic
property distributions, which confirms that the lattice retains the
boundary-dominated spectrum observed in the hyperbolic plane. We then
numerically investigate the response of the lattice to transient pulses
incident on the boundary and find that edge-confined wave propagation occurs.
The modal and transient pulse propagation behavior of the lattice is
experimentally validated in a milled aluminum sample. By leveraging hyperbolic
geometry, our mechanical lattice ushers in a novel class of mechanical
metamaterials with boundary-dominated wave phenomena reminiscent of
topologically protected systems suitable for applications in advanced wave
control.
