We have investigated characteristic solar wind dynamics associated with relativistic electron events at geosynchronous orbit. Most of the events for April, 1999 through December, 2002 are found to be accompanied by a prolonged solar quiet period which is characterized as low solar wind density, weak interplanetary magnetic field (IMF), and fast alfvenic fluctuations in IMF $B_z$. In a typical relativistic event, electron fluxes begin to increase by orders of magnitude when solar wind parameters drop to low values (e.g., $n_{sw}∼5 cm^{-3}$ and ｜$B_{IMF}$∼5 nT) after sharp peaks. Then the elevated electron fluxes stay at the high level during the solar quiet period. This observation may suggest the following scenario for the occurrence of a geosynchronous relativistic event: (ⅰ) Quiet solar winds can yield a stable and more dipole-like magnetospheric configurations in which the geosynchronous orbit locates well inside the trapping boundary of the energetic electrons. (ⅱ) If a large population of MeV electrons are generated (by whatever acceleration process(es)) in the inner magnetosphere, they can be trapped and effectively accumulated to a high intensity. (ⅲ) The high electron flux can persist for a number of days in the geosynchronous region as long as the solar wind dynamics stays quiet. Therefore the scenario indicates that the occurrence of a relativistic event would be a result of a delicate balance between the effects of electron acceleration and loss. In addition, the sensitive dependence of a relativistic event on the solar wind conditions makes the prediction of solar wind variability as important as understanding of electron acceleration processes in the forecast of a relativistic event.