  We have studied the detection of long-lived staus at the IceCube neutrino
telescope, after their production inside the Earth through the inelastic
scattering of high energy neutrinos. The theoretical predictions for the stau
flux are calculated in two scenarios in which the presence of long-lived staus
is naturally associated to viable supersymmetric dark matter. Namely, we
consider the cases with superWIMP (gravitino or axino) and neutralino dark
matter (along the coannihilation region). In both scenarios the maximum value
of the stau flux turns out to be about 1 event/yr in regions with a light stau.
This is consistent with light gravitinos, with masses constrained by an upper
limit which ranges from 0.2 to 15 GeV, depending on the stau mass. Likewise, it
is compatible with axinos with a mass of about 1 GeV and a very low reheating
temperature of order 100 GeV. In the case of the neutralino dark matter this
favours regions with a low value of tan(beta), for which the neutralino-stau
coannihilation region occurs for smaller values of the stau mass. Finally, we
study the case of a general supergravity theory and show how for specific
choices of non-universal soft parameters the predicted stau flux can increase
moderately.
