We construct a model structure on simplicial profinite sets such that the homotopy groups carry a natural profinite structure. This yields a rigid profinite completion functor for spaces and pro-spaces. One motivation is the étale homotopy theory of schemes in which higher profinite étale homotopy groups fit well with the étale fundamental group which is always profinite. We show that the profi...

Let f be a function such that for every ε > 0, nlog n ≤ f (n) ≤ nεn holds if n is sufficiently large. Suppose that log f (n)/ log n is nondecreasing. Using sequences of finite alternating groups, for every such f we construct a 4-generator group 0 such that sn(0), the number of subgroups of index at most n in 0, grows like f (n). This essentially completes the investigation of the “spectrum” of...

We solve an open problem of Herfort and Ribes: Profinite Frobenius groups of certain type do occur as closed subgroups of free profinite products of two profinite groups. This also solves a question of Pop about prosolvable subgroups of free profinite products.

For a profinite group, we construct a model structure on profinite spaces with a continuous action. We construct descent spectral sequences for the homotopy groups of the homotopy fixed point space and for the homology of homotopy orbit space which are strongly convergent for an arbitrary profinite group. Our main example is the Galois action on profinite étale topological types of schemes over...

In this paper, we show that any finitely generated abelian-byfinite group is an elementary submodel of its profinite completion. It follows that two finitely generated abelian-by-finite groups are elementarily equivalent if and only if they have the same finite images. We give an example of two finitely generated abelian-by-finite groups G, H which satisfy these properties while G x Z and H x Z...

This is a brief summary of one part of the forthcoming work “Cohomological control of deformation theory via A∞-structure.” Let G be a profinite group. The main result is that a natural A∞-structure on cohomology groups induces presentations of universal deformation rings for G-representations, more general moduli spaces for G-representations, and universal deformation rings for Galois pseudore...

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of F 2 × F 2 and we show that the profinite topology of the above group is strongly connected with the profinite topology of F 2 .

Some fundamental questions about infinite-vertex (free) profinite semigroupoids are clarified, putting in evidence differences with the finitevertex case. This is done with examples of free profinite semigroupoids generated by the graph of a subshift. It is also proved that for minimal subshifts, the infinite edges of such free profinite semigroupoids form a connected compact groupoid.

We investigate the continuous cohomology of infinite permutation groups on modules whose topology is profinite. To obtain acyclics we expand the class of modules to include those which are directed unions of their profinite submodules. As an application we give a criterion which implies finiteness of the continuous cohomology groups on finitely generated profinite modules for some familiar perm...