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In this paper we continue the study of algebraic fundamentale group ofminimal surfaces of general type S satisfying K_S^2<3\chi(S). We show that, ifK_S^2= 3\chi(S)-1 and the algebraic fundamental group of S has order 8, then Sis a Campedelli surface.  In view of the results of math.AG/0512483 and math.AG/0605733, this impliesthat the fundamental group of a surface with K^2<3\chi that has no irregularetale cover has order at most 9, and if it has order 8 or 9, then S is aCampedelli surface.  To obtain this result we establish some classification results for minimalsurfaces of general type such that K^2=3p_g-5 and such that the canonical mapis a birational morphism. We also study rational surfaces with a Z_2^3-action.

Surfaces with K^2<3\chi and finite fundamental group