  The intricate interplay between charge motion and magnetic order in
geometrically frustrated lattices is central for the properties of many
two-dimensional quantum materials. The triangular lattice antiferromagnet is a
canonical example of a frustrated system, and here we analyse the dynamics of a
hole in such a lattice focusing on observables that have become accessible in a
new generation of experiments. Using the $t$-$J$ model, we solve the problem
exactly within linear spin wave theory in the limit of strong magnetic
interactions, showing that the ground state is described by a coherent state of
spin waves. The derivation highlights the crucial role played by the
interaction between a static hole and the neighboring spins, which originates
in the geometric frustration and has often been omitted in earlier works.
Furthermore, we show that the non-equilibrium dynamics after a hole has
abruptly been inserted at a lattice site is given exactly by a coherent state
with time-dependent oscillatory coefficients. Physically, this describes a
burst of magnetic frustration propagating through only two-thirds of the
lattice sites, since a destructive interference of spin waves leaves spins
parallel to that removed by the hole unperturbed. After the wave has propagated
through the lattice, the magnetization relaxes to that of the ground state. We
then use our analytical solution to benchmark the widely used self-consistent
Born approximation (SCBA), showing that it is very accurate also for a
triangular lattice. The magnetic polaron spectrum is analysed for general
magnetic interactions using the SCBA, and we compare our results with those for
a square lattice.
