Homology and closure properties of autostackable groups

Autostackability for finitely presented groups is a topological property of the Cayley graph combined with formal language theoretic restrictions, that implies solvability of the word problem. The class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix-closed normal form set, and all groups admitting finite complete rewriting systems. Although groups in the latter two classes all satisfy the homological finiteness condition $FP_\infty$, we show that the class of autostackable groups includes a group that is not of type $FP_3$. We also show that the class of autostackable groups is closed under graph products and extensions.Comment: 20 page

Reduced Steenrod operations and resolution of singularities

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.Comment: Final version, to appear in J. K-theor

Automorphisms of local fields of period pp and nilpotent class <p<p

Suppose $K$ is a finite field extension of $\mathbb{Q} _p$ containing a primitive $p$-th root of unity. Let $\Gamma _{<p}$ be the Galois group of a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In the paper we describe the ramification filtration $\{\Gamma _{<p}^{(v)}\}_{v\geqslant 0}$ and relate it to an explicit form of the Demushkin relation for $\Gamma _{<p}$. The results are given in terms of Lie algebras attached to involved groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.Comment: Substantial revision, 61 page