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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.

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)
Full Text: DOI arXiv
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