In previous work, the first author established a natural bijection between minimal subshifts and maximal regular J -classes of free profinite semigroups. In this paper, the Schützenberger groups of such J -classes are investigated, in particular in respect to a conjecture proposed by the first author concerning their profinite presentation. The conjecture is established for all non-periodic min...

It is known that Seifert-van Kampen theorem (for “good” topological spaces) can be showed by arguing the category of covering spaces. Similar arguments should be valid for abstract Galois categories (which Mochizuki calls ”connected anabelioids”) and neutral Tannakian categories. But when we try to state the theorem, the problem is the existence of amalgams (in other words, fibered coproducts) ...

γ = c0 + c1p+ c2p + · · · = (. . . c3c2c1c0)p, with ci ∈ Z, 0 ≤ ci ≤ p− 1, called the digits of γ. This ring has a topology given by a restriction of the product topology—we will see this below. The ring Zp can be viewed as Z/pZ for an ‘infinitely high’ power n. This is a useful idea, for example, in the study of Diophantine equations: if such an equation has a solution in the integers, then it...

We consider finitely presented, residually finite groups G and finitely generated normal subgroups A such that the inclusion A ↪→ G induces an isomorphism from the profinite completion of A to a direct factor of the profinite completion of G. We explain why A need not be a direct factor of a subgroup of finite index in G; indeed G need not have a subgroup of finite index that splits as a non-tr...