  Symmetric couplings among aggregates of $n$ chromophores increase the
transfer rate of excitons by a factor $n^2$, a quantum mechanical phenomenon
called "supertransfer." In this work we demonstrate how supertransfer effects
induced by geometrical symmetries can enhance the exciton diffusion length by a
factor $n$ along cylindrically symmetric structures, consisting of arrays of
rings of chromophores, and along spiral arrays. We analyse both closed system
dynamics and open quantum dynamics, modelled by combining a random bosonic bath
with static disorder. In the closed system case, we use the symmetries of the
system within a short-time approximation to obtain a closed analytical
expression for the diffusion length that explicitly reveals the supertransfer
contribution. When subject to disorder, we show that supertransfer can enhance
excitonic diffusion lengths for small disorders and characterize the crossover
from coherent to incoherent motion. Owing to the quasi-1D nature of the model,
disorder ultimately localizes the excitons, diminishing but not destroying the
effects of supertransfer. When dephasing effects are included, we study the
scaling of diffusion with both time and number of chromophores and observe that
the transition from a coherent, ballistic regime to an incoherent, random-walk
regime occurs at the same point as the change from supertransfer to classical
scaling.
