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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)
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