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The main result of this paper is the following theorem. Let q be a prime, A an elementary abelian group of order q^3. Suppose that A acts as a coprime group of automorphisms on a profinite group G in such a manner that C_G(a)' is periodic for each nontrivial element a in A. Then G' is locally finite.

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based profinite group as a point in a Polish space. Then we study the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs.

We obtain some general restrictions on the continuous endomorphisms of a profinite group G under the assumption that G has only finitely many open subgroups of each index (an assumption which automatically holds, for instance, if G is finitely generated). In particular, given such a group G and a continuous endomorphism phi we obtain a semidirect decomposition of G into a 'contracting' normal subgroup and a complement on which phi induces an automorphism; both the normal subgroup and the complement are closed. If G is isomorphic to a proper open subgroup of itself, we show that G has an infinite abelian normal pro-p subgroup.

Let $G$ be a profinite group with a countable basis of neighborhoods of the identity. A cohomology and homology theory for the group $G$ with non-discrete topological coefficients is developed, improving previous expositions of the subject. Though the category of topological $G$-modules considered is additive but not abelian, there is a theory of derived functors. All standard properties of group cohomology and homology are then obtained rephrasing the standard proofs given in the abelian categories' setting. In this way, one gets the universal coefficients Theorem, Lyndon/Hochschild-Serre spectral sequence and Shapiro's Lemma. Another interesting feature of this theory is that it allows to rephrase and easily prove for profinite groups, all definitions and results about cohomological dimension and duality which hold for discrete groups. No claim is made on the originality of the results here exposed but rather on the presentation of the subject.

A profinite group G is just infinite if every non-trivial closed normal subgroup of G is of finite index, and hereditarily just infinite if every open subgroup is just infinite. Hereditarily just infinite profinite groups need not be virtually pro-p, as shown in a recent paper of Wilson. The same paper gives a criterion on an inverse system of finite groups that is sufficient to ensure the limit is either virtually abelian or hereditarily just infinite. We give criteria of a similar nature that characterise the just infinite and hereditarily just infinite properties under the assumption that G is not virtually pro-p.

For every profinite group $G$, we construct two covariant functors $\Delta_G$ and ${\bf {\mathcal {AP}}}_G$ from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor $W_G$ introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, {\it Adv. in Math.} {\bf{70}} (1988), 87-132]. We call $\Delta_G$ the generalized Burnside-Grothendieck ring functor and ${\bf {\mathcal {AP}}}_G$ the aperiodic ring functor (associated with $G$). In case $G$ is abelian, we also construct another functor ${\bf Ap}_G$ from the category of commutative rings with identity to itself as a generalization of the functor ${\bf Ap}$ introduced in [K. Varadarajan, K. Wehrhahn, Aperiodic rings, necklace rings, and Witt vectors, {\it Adv. in Math.} {\bf 81} (1990), 1-29]. Finally it is shown that there exist $q$-analogues of these functors (i.e, $W_G, \Delta_G, {\bf {\mathcal {AP}}}_G$, and ${\bf Ap}_G$) in case $G=\hat C$ the profinite completion of the multiplicative infinite cyclic group.

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the given automorphisms. (2) Given a natural number $q$, there exist $C=C(q)$ and $M=M(q)$ such that: if $S$ is a finite quasisimple group with $| S/\mathrm{Z}(S)| >C$, $\beta_{j}$ $ (j=1,...,M)$ are any automorphisms of $S$, and $q_{j}$ $ (j=1,...,M)$ are any divisors of $q$, then there exist inner automorphisms $\alpha_{j}$ of $S$ such that $S=\prod_{1}^{M}[S,(\alpha_{j}\beta_{j})^{q_{j}}]$. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.

We give a definition of weakly sofic groups (w-sofic groups). Our definition is rather natural extension of the definition of sofic groups where instead of Hamming metric on symmetric groups we use general bi-invariant metrics on finite groups. The existence of non w-sofic groups is equivalent to some conjecture about profinite topology on free groups.

We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some monoid. This is a generalization of the fact that the topos associated to the \'etale site of a scheme is equivalent to the category of sets with continuous action by the \'etale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.

For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $\gamma_k$-values in $G$, it is shown that $\gamma_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contains all $\gamma_k$-values, it is shown that $\gamma_k(G)$ has finite $(k,r,s)$-bounded rank.

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type $\operatorname{FP}_n$ over a profinite ring $R$, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type $\operatorname{FP}_n$ is closed under extensions, quotients by subgroups of type $\operatorname{FP}_n$, proper amalgamated free products and proper $\operatorname{HNN}$-extensions, for each $n$. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type $\operatorname{FP}_\infty$ over all profinite $R$. For any class $\mathcal{C}$ of finite groups closed under subgroups, quotients and extensions, we also construct pro-$\mathcal{C}$ groups of type $\operatorname{FP}_n$ but not of type $\operatorname{FP}_{n+1}$ over $\mathbb{Z}_{\hat{\mathcal{C}}}$ for each $n$. Finally, we show that the natural analogue of the usual condition measuring when pro-$p$ groups are of type $\operatorname{FP}_n$ fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.

Let $G$ be a residually finite group. To any decreasing sequence $\mathcal S = (H_n)_n $ of finite index subgroups of $G$ is associated a unitary representation $\rho_{\mathcal S}$ of $G$ in the Hilbert space $\bigoplus_{n=0}^{+\infty} \ell^2 (G/H_n) $. This paper investigates the following question: when does the representation $\rho_{\mathcal S} $ weakly contain the regular representation $\lambda$ of $G$?

Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\Emb(S)$ has finitely many isomorphism classes. For instance, we give an example where $\Emb(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\Emb(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.

The set of all closed subgroups of a profinite carries a natural profinite topology. This space of subgroups can be classified up to homeomorphism in many cases, and tight bounds placed on its complexity as expressed by its scattered height.

Assume $G$ is a polycyclic group and $\phi:G\to G$ an endomorphism. Let $G\ast_{\phi}$ be the ascending HNN extension of $G$ with respect to $\phi$; that is, $G\ast_{\phi}$ is given by the presentation $$G\ast_{\phi}= < G, t \ |\ t^{-1}gt = \phi(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_{\phi}}$ be the profinite completion of $G\ast_{\phi}$. We prove that, for any finite discrete $\hat{G\ast_{\phi}}$-module $A$, the map $H^*(\hat{G\ast_{\phi}}, A)\to H^*(G\ast_{\phi},A)$ induced by the canonical map $G\ast_{\phi}\to \hat{G\ast_{\phi}}$ is an isomorphism.

We define the profinite completion of a C*-algebra, which is a pro-C*-algebra, as well as the pro-C*-algebra of a profinite group. We show that the continuous representations of the pro-C*-algebra of a profinite group correspond to the unitary representations of the group which factor through a finite group. We define natural homomorphisms from the C*-algebra of a locally compact group and its profinite completion to the pro-C*-algebra of the profinite completion of the group. We give some conditions for injectivity or surjectivity of these homomorphisms, but an important question remains open.

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on $\mathbb Z_p$.

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a profinite group G, where k is a finite field of characteristic p. We define the concept of relative projectivity for a profinite k[[G]]-module. We prove a characterization of finitely generated relatively projective modules analogous to the finite case with additions of interest to the profinite theory. We introduce vertices and sources for indecomposable finitely generated k[[G]]-modules and show that the expected conjugacy properties hold - for sources this requires additional assumptions. Finally we prove a direct analogue of Green's Indecomposability Theorem for finitely generated modules over a virtually pro-p group.

We consider profinite groups in which all commutators are contained in a union of finitely many procyclic subgroups. It is shown that if G is a profinite group in which all commutators are covered by m procyclic subgroups, then G possesses a finite characteristic subgroup M contained in G' such that the order of M is m-bounded and G'/M is procyclic. If G is a pro-p group such that all commutators in G are covered by m procyclic subgroups, then G' is either finite of m-bounded order or procyclic.

We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.

We study a generalisation of the family of non-(virtually pro-$p$) hereditarily just infinite profinite groups introduced by J.\! S.\! Wilson in 2010. We prove that this family contains groups of finite lower rank. We also show that many groups in this family are not topologically finitely presentable.

The sets of closed and closed-normal subgroups of a profinite group carry a natural profinite topology. Through a combination of algebraic and topological methods the size of these subgroup spaces is calculated, and the spaces partially classified up to homeomorphism.

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.

Let $g$ be an element of a group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$ over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$, we prove that if, for some $n$, $|E_n(g)|\leq m$ for all $g\in G$, then the order of the nilpotent residual $\gamma _{\infty}(G)$ is bounded in terms of $m$.

Let $G$ be a profinite group in which for every element $x\in G$ there exists a natural number $q=q(x)$ such that $x^q$ is Engel. We show that $G$ is locally virtually nilpotent. Further, let $p$ be a prime and $G$ a finitely generated profinite group in which for every $\gamma_k$-value $x\in G$ there exists a natural $p$-power $q=q(x)$ such that $x^q$ is Engel. We show that $\gamma_k(G)$ is locally virtually nilpotent.

We consider the p-Zassenhaus filtration (G_n) of a profinite group G. Suppose that G=S/N for a free profinite group S and a normal subgroup N of S contained in S_n. Under a cohomological assumption on the n-fold Massey products (which holds e.g., if the p-cohomological dimension of G is at most 1), we prove that G_{n+1} is the intersection of all kernels of upper-triangular unipotent (n+1)-dimensional representations of G over \mathbb F_p. This extends earlier results by Minac, Spira, and the author on the structure of absolute Galois groups of fields.

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers.

Broadly speaking, a finiteness property of groups is any generalisation of the property of having finite order. A large part of infinite group theory is concerned with finiteness properties and the relationships between them. Profinite groups are an important case of this, being compact topological groups that possess an intimate connection with their finite images. This thesis investigates the relationship between several finiteness properties that a profinite group may have, with consequences for the structure of finite and profinite groups.

Let R be any ring (with 1), \Gamma a group and R\Gamma the corresponding group ring. Let H be a subgroup of \Gamma of finite index. Let M be an R\Gamma -module, whose restriction to RH is projective. Moore's conjecture: Assume for every nontrivial element x in \Gamma, at least one of the following two conditions holds: M1) the subgroup generated by x intersects H non-trivially (in particular this holds if \Gamma is torsion free). M2) ord(x) is finite and invertible in R. Then M is projective as an R\Gamma-module. More generally, the conjecture has been formulated for crossed products R*\Gamma and even for strongly graded rings R(\Gamma). We prove the conjecture for new families of groups, in particular for groups whose profinite completion is torsion free. The conjecture can be formulated for profinite modules M over complete groups rings [[R\Gamma ]] where R is a profinite ring and \Gamma a profinite group. We prove the conjecture for arbitrary profinite groups. This implies Serre's theorem on cohomological dimension of profinite groups.

In this note, we explore the notion of hyperbolicity of finitely and infinitely generated groups in case of profinite groups. Some applications to Diophantine geometry are suggested. In particular, we reformulate certain problems in Diophantine geometry in term of hyperbolic profinite groups.

We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the $\J$-class associated to an irreducible sofic shift.

We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow \Gamma$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehat{\Gamma}$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehat{\Gamma}$.

We define and study the class of positively finitely related (PFR) profinite groups. Positive finite relatedness is a probabilistic property of profinite groups which provides a first step to defining higher finiteness properties of profinite groups which generalize the positively finitely generated groups introduced by Avinoam Mann. We prove many asymptotic characterisations of PFR groups, for instance we show the following: a finitely presented profinite group is PFR if and only if it has at most exponential representation growth, uniformly over finite fields (in other words: the completed group algebra has polynomial maximal ideal growth). From these characterisations we deduce several structural results on PFR profinite groups.

We prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general.

We study the Hausdorff dimension of R-analytic subgroups in an R-analytic profinite group, where R is a pro-p ring whose asso- ciated graded ring is an integral domain. In particular, we prove that the set of such Hausdorff dimensions is a finite subset of the rational numbers.

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C_1,...,C_n and the minimal number of generators of G is n.

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an FG-module, then the dimension of the second cohomology group of G with coefficients in M is at most 17.5 times the dimension of M. The third result is that we may replace 17.5 by 18.5 as long as M is faithful irreducible G-module. These last two results answer conjectures of Holt.

Let $w$ be a group-word. Suppose that the set of all $w$-values in a profinite group $G$ is contained in a union of countably many subgroups. It is natural to ask in what way the structure of the verbal subgroup $w(G)$ depends on the properties of the covering subgroups. The present article is a survey of recent results related to that question. In particular we survey results on finite and countable coverings of word-values (mostly commutators) by procyclic, abelian, nilpotent, and soluble subgroups, as well as subgroups with finiteness conditions. The last section of the paper is devoted to relation of the described results with Hall's problem on conciseness of group-words.

$p$-Adic compactifications of geometric loop and diffeomorphism groups of compact manifolds on finite-dimensional spaces over non-Archimedean fields are investigated. Weakened topology is introduced. The structure of newly constructed compact groups is studied. Representations of such profinite groups are discused.

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let $w$ be a `locally finite' group word and $d\in\mathbb{N}$. Then there exists $f=f(w,d)$ such that in every $d$-generator finite group $G$, every element of the verbal subgroup $w(G)$ is equal to a product of $f$ $w$-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.

A profinite group $G$ is just infinite if every closed normal subgroup of $G$ is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup $H$ of $G$, there are only finitely many open normal subgroups of $G$ not contained in $H$. This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola, who proved the same characterisation in the case of pro-$p$ groups. We also use this result to establish a number of features of the general structure of profinite groups with regard to the just infinite property.

Inverse limits and profinite groups are used in a quantum mechanical context. Two cases are considered. A quantum system with positions in the profinite group ${\mathbb Z}_p$ and momenta in the group ${\mathbb Q}_p/{\mathbb Z}_p$; and a quantum system with positions in the profinite group ${\hat {\mathbb Z}}$ and momenta in the group ${\mathbb Q}/{\mathbb Z}$. The corresponding Schwatz-Bruhat spaces of wavefunctions and the Heisenberg-Weyl groups are discussed. The sets of subsystems of these systems are studied from the point of view of partial order theory. It is shown that they are directed-complete partial orders. It is also shown that they are topological spaces with $T_0$ topologies, and this is used to define continuity of various physical quantities. The physical meaning of profinite groups, non-Archimedean metrics, partial orders and $T_0$ topologies, in a quantum mechanical context, is discussed.

We study simplicial profinite groups with a view towards applications in profinite combinatorial group theory. This approach provides a natural framework to the concept of pro-$\mathfrak{C}$-presentation of a pro-$\mathfrak{C}$-group $G$ as a 1-truncation of its free simplicial pro-$\mathfrak{C}$-resolution. The category of simplicial pro-$\mathfrak{C}$-groups has a closed simplicial model category structure. This yields a possibility to define some old and new derived functors as left Quillen derived functors from this simplicial model category. When $\mathfrak{C}$ is L-groups, than may construct free simplical pro-L-resolution functorially. We introduce settings of $\Delta$-adic and Zassenhaus filtrations for free simplical pro-p-resolutions and derive some calculations for pro-p-groups. The usage of pro-p-Curtis-Rector spectral sequences sheds homotopical light on Golod-Shafarevitch type results.

The main result of the paper is the following theorem. Let $q$ be a prime and $A$ an elementary abelian group of order $q^3$. Suppose that $A$ acts coprimely on a profinite group $G$ and assume that $C_G(a)$ is locally nilpotent for each $a\in A^{\#}$. Then the group $G$ is locally nilpotent.

We initiate the study of profinite groups of non-negative deficiency. The principal focus of the paper is to show that the existence of a finitely generated normal subgroup of infinite index in a profinite group $G$ of non-negative deficiency gives rather strong consequences for the structure of $G$. To make this precise we introduce the notion of $p$-deficiency ($p$ a prime) for a profinite group $G$. This concept is more useful in the study of profinite groups then the notion of deficiency. We prove that if the $p$-deficiency of $G$ is positive and $N$ is a finitely generated normal subgroup such that the $p$-Sylow subgroup of $G/N$ is infinite and $p$ divides the order of $N$ then we have $\cd_p(G)=2$, $\cd_p(N)=1$ and $\vcd_p(G/N)=1$ for the cohomological $p$-dimensions; moreover either the $p$-Sylow subgroup of $G/N$ is virtually cyclic or the $p$-Sylow subgroup of $N$ is cyclic. A profinite Poincar\'e duality group $G$ of dimension 3 at a prime $p$ ($PD^3$-group) has deficiency 0. In this case we show that for $N$ and $p$ as above either $N$ is $PD^1$ at $p$ and $G/N$ is virtually $PD^2$ at $p$ or $N$ is $PD^2$ at $p$ and $G/N$ is virtually $PD^1$ at $p$. In particular if $G$ is pro-$p$ then either $N$ is infinite cyclic and $G/N$ is virtually Demushkin or $N$ is Demushkin and $G/N$ is virtually infinite cyclic. We apply this results to deduce structural information on the profinite completions of ascending HNN-extensions of free groups. We also give some implications of our theory to the congruence kernels of certain arithmetic groups.

We show that the class of profinite duality groups is closed under group extensions provided that the kernel satisfies some finiteness condition. This extends earlier results of Pletch and of Wingberg.

We will investigate quasi-randomness for profinite groups. We will obtain bounds for the mininal degree of non-trivial representations of $\SL_k(\Z{p^n})$ and $\Sp_{2k}(\Z{p^n})$. Our method also delivers a lower bound for the minimal degree of a faithful representation for these groups. Using the suitable machinery from functional analysis, we establish exponential lower and upper bounds for the supremal measure of a product-free measurable subset of the profinite groups $\SL_{k}({\ZZ_p})$ and $\Sp_{2k}(\ZZ_p)$. We also obtain analogous bounds for a special subgroup of the automorphism group of a regular tree.

For any positive integer $n$, $\mathcal{A}_n$ is the class of all groups $G$ such that, for $0\leq i\leq n$, $H^i(\hat{G},A)\cong H^i(G,A)$ for every finite discrete $\hat{G}$-module $A$. We describe certain types of free products with amalgam and HNN extensions that are in some of the classes $\mathcal{A}_n$. In addition, we investigate the residually finite groups in the class $\mathcal{A}_2$.

We classify the hulls of different limit-periodic potentials and show that the hull of a limit-periodic potential is a procyclic group. We describe how limit-periodic potentials can be generated from a procyclic group and answer arising questions. As an expository paper, we discuss the connection between limit-periodic potentials and profinite groups as completely as possible and review some recent results on Schroedinger operators obtained in this context.

Consider a free ergodic measure preserving profinite action $\Gamma\curvearrowright X$ (i.e. an inverse limit of actions $\Gamma\curvearrowright X_n$, with $X_n$ finite) of a countable property (T) group $\Gamma$ (more generally of a group $\Gamma$ which admits an infinite normal subgroup $\Gamma_0$ such that the inclusion $\Gamma_0\subset\Gamma$ has relative property (T) and $\Gamma/\Gamma_0$ is finitely generated) on a standard probability space $X$. We prove that if $w:\Gamma\times X\to \Lambda$ is a measurable cocycle with values in a countable group $\Lambda$, then $w$ is cohomologous to a cocycle $w'$ which factors through the map $\Gamma\times X\to \Gamma\times X_n$, for some $n$. As a corollary, we show that any orbit equivalence of $\Gamma\curvearrowright X$ with any free ergodic measure preserving action $\Lambda\curvearrowright Y$ comes from a (virtual) conjugacy of actions.