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Subgroup structure of fundamental groups in positive characteristic. (English) Zbl 1276.14044
Let $$C$$ be a smooth affine curve over an algebraically closed field $$k$$ of characteristic $$p>0$$. The structure of its étale fundamental group $$\pi_1^{\text{ét}}(C)$$ is still mysterious, even though certain quotients are well understood. For example one can describe the prime-to-$$p$$ part of $$\pi^{\text{ét}}_1(C)$$, and it is known that the maximal pro-$$p$$-quotient is a free pro-$$p$$-group on $$\# k$$ generators.
The conjecture of Abhyankar, proved by M. Raynaud for the affine line [Invent. Math. 116, No. 1–3, 425–462 (1994; Zbl 0798.14013)], and by D. Harbater in general [Invent. Math. 117, No. 1, 1–25 (1994; Zbl 0805.14014)], precisely describes which finite groups occur as quotients of $$\pi_1^{\text{ét}}(C)$$; but even this is not sufficient to determine the group $$\pi_1^{\text{ét}}(C)$$.
The present article sheds some light on the structure of $$\pi_1^{\text{ét}}(C)$$ by studying certain subgroups. To describe the main result, we continue to omit base points from the notation. If $$U\subset\pi_1^{\text{ét}}(C)$$ is an open normal subgroup, then there exists a finite étale covering $$Z\rightarrow C$$, such that $$U=\pi_1^{\text{ét}}(Z)$$. For an integer $$g\geq 0$$, denote by $$P_g(C)$$ the intersection of all open normal subgroups $$\pi_1^{\text{ét}}(Z)\subset\pi_1^{\text{ét}}(C)$$, such that the smooth compactification of the curve $$Z$$ has genus $$\geq g$$.
Assume that the base field $$k$$ is countable. Let $$M\subset \pi_1^{\text{ét}}(C)$$ be a closed subgroup with $$M\subset P_g(C)$$ for some $$g\geq 0$$. The main result of the article under review is a so-called “diamond criterion” for $$M$$ to be profinite free. In particular, it implies that:
1) The commutator subgroup of $$\pi_1^{\text{ét}}(C)$$ is profinite free (this was previously proved by the second author [M. Kumar, J. Algebra 319, No. 12, 5178–5207 (2008; Zbl 1197.14026)]).
2) The groups $$P_g(C)$$ are profinite free.
3) If $$N$$ is a closed normal subgroup of infinite index, and $$M$$ a proper open subgroup of $$N$$, which is contained in $$P_g(C)$$ for some $$g$$, then $$M$$ is profinite free.
The proof of the main theorem relies on a geometric construction which shows that certain embedding problems for $$\pi_1^{\text{ét}}(C)$$ are solvable.

##### MSC:
 14H30 Coverings of curves, fundamental group 14G17 Positive characteristic ground fields in algebraic geometry 14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
##### Keywords:
fundamental groups; free groups; coverings; diamond theorems
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##### References:
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