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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 groupalg 1 (S): • Assume that K 2 S  3(S). Then S has an irregularcover if and only if S has a free pencil of hyperelliptic curves of genus 3 with at least 4 double fibres. • If K 2 S = 3 and (S) = 1, then S has no irregularcover. • If K 2 < 3(S) and S does not have any irregularcover, then | alg (S)|  9. If | alg (S)| = 9, then K 2 S = 2, (S) = 1.

On the algebraic fundamental group of surfaces with $K^2≤3x$