Group theory and semigroup theory have developed in somewhat different directions in the past several decades. While Cayley’s theorem enables us to view groups as groups of permutations of some set, the analogous result in semigroup theory represents semigroups as semigroups of functions from a set to itself. Of course both group theory and semigroup theory have developed significantly beyond t...

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...

We study profinite completion of spaces in the model category of profinite spaces and construct a rigidification of the completion functors of Artin-Mazur and Sullivan which extends also to non-connected spaces. Another new aspect is an equivariant profinite completion functor and equivariant fibrant replacement functor for a profinite group acting on a space. This is crucial for applications w...

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...

We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if ...

It is well known that clopen subgroups of finitely generated free profinite groups are again finitely generated free profinite groups. Clopen submonoids of free profinite monoids need not be finitely generated nor free. Margolis, Sapir and Weil proved that the closed submonoid generated by a finite code (which is in fact clopen) is a free profinite monoid generated by that code. In this note we...

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 cha...