arXiv subjects
Publications and source records attributed to Andrey Mikhovich.
Quasirational (pro-$p$)presentations are studied. The difference between aspherical and quasirational presentations sheds some light on the Whitehead's conjecture. We confirm expectations O.V. Melnikov on existence of a proper class of presentations.
We prove the Identity Theorem for pro-$p$-groups with a single defining relation giving a positive feedback to a question of Serre on the structure of relation modules. A construction of "conjurings" indicates finality of our result in a certain sense.
Quasirational presentations ($QR$-presentations) of (pro-$p$)groups are studied. Such presentations include, in particular, aspherical presentations of discrete groups and their subpresentations and pro-$p$-groups with a single defining relation. Using schematization of $QR$-presentations we answer a variation of J.-P. Serre's question on one-relator pro-$p$-groups.
We introduce the concept of quasirational relation modules for discrete and pro-$p$ presentations of discrete and pro-$p$ groups and show that aspherical presentations and their subpresentations are quasirational. In the pro-$p$-case quasirationality of pro-$p$-groups with a single defining relation holds. For every quasirational (pro-$p$)relation module we construct the so called $p$-adic rationalization, which is a pro-fd-module $\overline{R}\widehat{\otimes}\mathbb{Q}_p= \varprojlim R/[R,R\mathcal{M}_n]\otimes\mathbb{Q}_p$. We provide the isomorphisms $\overline{R^{\wedge}_w}(\mathbb{Q}_p)=\overline{R}\widehat{\otimes}\mathbb{Q}_p$ and $\overline{R_u}(\mathbb{Q}_p)=\mathcal{O}(G_u)^*$, where $R^{\wedge}_w$ and $R^{\wedge}_u$ stands for continuous prounipotent completions and corresponding prounipotent presentations correspondingly. We show how $\overline{R^{\wedge}_{w}}$ embeds into a sequence of abelian prounipotent groups. This sequence arises naturally from a certain prounipotent crossed module, the latter bring concrete examples of proalgebraic homotopy types. The old-standing open problem of Serre, slightly corrected by Gildenhuys, in its modern form states that pro-$p$-groups with a single defining relation are aspherical. Our results give a positive feedback to the question of Serre.
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 $Δ$-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.
We introduce the concept of quasirational relation modules for discrete (pro- p) presentations of discrete (pro-p) groups. It is shown, that this class of presentations for discrete groups contains CA-presentations and their subpresentations. For pro-p-groups we see that all presentations of pro-p-groups with a single defining relation are quasirational. We offer definitions of p- adic G(p)-completion and p-adic rationalization of relation modules which are adjusted to quasirational pro-p-presentations. p-adic rationalizations of quasirational relation modules of pro-p-groups are isomorphic to abelianized p-adic Malcev completions.