On the algebraic fundamental group of surfaces with K^2\leq 3χ
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Let S be a minimal complex surface of general type with $q(S)=0$. We prove the following statements concerning the algebraic fundamental group:
I) Assume that K^2_S\leq 3χ(S). Then S has an irregular etale cover if and only if S has a free pencil of hyperelliptic curves of genus 3 with at least 4 double fibres. II) If K^2_S=3 and χ(S)=1, then S has no irregular etale cover.
III) If K^2_S<3χ(S) and S does not have any irregular etale cover, then the order of the algebraic fundamental group is lesser or equal to 9, and if equality occurs then K^2_S=2,
χ(S)=1.
Final version, to appear in J.D.G
Final version, to appear in J.D.G