There has been much recent interest into those properties of a 3manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-p completion of certain graph manifold groups. Z = lim ← − Z/n of all finite cyclic groups. The pro-π completion, denoted Z π , is the inverse limit of those finite cyclic groups whose orders only involve primes from π. Because of the Chinese Remainder Theorem, these finite cyclic groups split as products of cyclic groups of prime power order, and these splittings are natural with respect to the quotient maps Z/mn → Z/n. It follows that the profinite completion of Z splits as the direct product, over all primes p, of the rings of p-adic integersThis splitting as a direct product, which of course is not a feature of Z itself, means that ring-theoreticallyẐ behaves less well than Z. In particular, there

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all Seifert fibre spaces are distinguished from each other by their profinite completions apart from some previously-known examples due to Hempel. We also characterize when bounded Seifert fibre space groups have isomorphic profinite completions, given some conditions on the boundary.

In this paper we define and develop the theory of the cohomology of a profinite group relative to a collection of closed subgroups. Having made the relevant definitions we establish a robust theory of cup products and use this theory to define profinite Poincaré duality pairs. We use the theory of groups acting on profinite trees to give Mayer-Vietoris sequences, and apply this to give results concerning decompositions of 3-manifold groups. Finally we discuss the relationship between discrete duality pairs and profinite duality pairs, culminating in the result that profinite completion of the fundamental group of a compact aspherical 3-manifold is a profinite Poincaré duality group relative to the profinite completions of the fundamental groups of its boundary components.• the restriction of T to C π pGq commutes with inverse limits in the sense thatOne defines notions of connected sequences of continuous functors, and continuous cohomological and homological functors in the natural way. A continuous cohomological functor T ‚ is co-effaceable if T i pJq " 0 for all i ą 0 and all injective modules J P D π pGq. Similarly for effaceability.

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We classify pro-p Poincaré duality pairs in dimension two. We then use this classification to build a pro-p analogue of the curve complex and establish its basic properties. We conclude with some statements concerning separability properties of the mapping class group.