On ℓ-adic Pro-algebraic and Relative Pro-ℓ Fundamental Groups

We recall ℓ-adic relative Malcev completions and relative pro-ℓ completions of pro-finite groups and homotopy types. These arise when studying unipotent completions of fibres or of normal subgroups. Several new properties are then established, relating to ℓ-adic analytic moduli and comparisons between relative Malcev and relative pro-ℓ completions. We then summarise known properties of Galois actions on the pro-ℚℓ-algebraic geometric fundamental group and its big Malcev completions. For smooth varieties in finite characteristics different from ℓ, these groups are determined as Galois representations by cohomology of semisimple local systems. Olsson’s non-abelian étale-crystalline comparison theorem gives slightly weaker results for varieties over ℓ-adic fields, since the non-abelian Hodge filtration cannot be recovered from cohomology.
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