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We consider a point process sequence induced by a stationary symmetricalpha-stable (0 < alpha < 2) discrete parameter random field. It is easy toprove, following the arguments in the one-dimensional case in Resnick andSamorodnitsky (2004), that if the random field is generated by a dissipativegroup action then the point process sequence converges weakly to a clusterPoisson process. For the conservative case, no general result is known even inthe one-dimensional case. We look at a specific class of stable random fieldsgenerated by conservative actions whose effective dimensions can be computedusing the structure theorem of finitely generated abelian groups. Thecorresponding point processes sequence is not tight and hence needs to beproperly normalized in order to ensure weak convergence. This weak limit iscomputed using extreme value theory and some counting techniques.

Ergodic theory, Abelian groups and point processes induced by stable random fields