Lepton Private Higgs and the discrete group \Sigma(81)

We use the discrete group \Sigma(81) = (Z_3 x Z_3 x Z_3)\rtimes Z_3 to explore a particular region of parameter space in the Private Higgs model. In doing so we suggest a relation among the off-diagonal entries of the neutrino mass matrix and a possible explanation for the muon magnetic moment anomaly, a_\mu^{exp}-a_\mu^{SM} ~ 10^{-9}. We predict three new nearly degenerate Higgs doublets with masses of order ~ 500 GeV to ~ 1 TeV, and three nearly degenerate SM-singlet TeV-scale neutrinos. The largest scale in the model is ~ 10 TeV, so there is no severe hierarchy problem. The appendix is devoted to the group theory of \Sigma(81).