arXiv subjects

Roman Mikhailov

Publications and source records attributed to Roman Mikhailov.

60 records · Page 4Linked to original sources

arXiv2008

A limit approach to group homology

In this paper, we consider for any free presentation $G = F/R$ of a group $G$ the coinvariance $H_{0}(G,R_{ab}^{\otimes n})$ of the $n$-th tensor power of the relation module $R_{ab}$ and show that the homology group $H_{2n}(G,{\mathbb Z})$ may be identified with the limit of the groups $H_{0}(G,R_{ab}^{\otimes n})$, where the limit is taken over the category of these presentations of $G$. We also consider the free Lie ring generated by the relation module $R_{ab}$, in order to relate the limit of the groups $γ_{n}R/[γ_{n}R,F]$ to the $n$-torsion subgroup of $H_{2n}(G,{\mathbb Z})$.

math.GR↗

arXiv2008

Free subalgebras of Lie algebras close to nilpotent

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n\geq 1, let L_{n+2} be a Lie algebra with generator set x_1,..., x_{n+2} and the following relations: for k\leq n, any commutator of length $k$ which consists of fewer than k different symbols from {x_1,...,x_{n+2}} is zero. As an application of this result about automata algebras, we prove that for every n\geq 1, L_{n+2} contains a free subalgebra. We also prove the similar result about groups defined by commutator relations.

math.RA↗

arXiv2008

A colimit of classifying spaces

We recall a group-theoretic description of the first non-vanishing homotopy group of a certain (n+1)-ad of spaces and show how it yields several formulae for homotopy and homology groups of specific spaces. In particular we obtain an alternative proof of J. Wu's group-theoretic description of the homotopy groups of a 2-sphere.

math.GR↗

arXiv2008

Intersection of subgroups in free groups and homotopy groups

We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $π_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the computation of $π_2$. Let $K$ be a two-dimensional CW-complex with subcomplexes $K_1,K_2,K_3$ such that $K=K_1\cup K_2\cup K_3$ and $K_1\cap K_2\cap K_3$ is the 1-skeleton $K^1$ of $K$. We construct a natural homomorphism of $π_1(K)$-modules $$ π_3(K)\to \frac{R_1\cap R_2\cap R_3}{[R_1,R_2\cap R_3][R_2,R_3\cap R_1][R_3,R_1\cap R_2]}, $$ where $R_i=ker\{π_1(K^1)\to π_1(K_i)\}, i=1,2,3$ and the action of $π_1(K)=F/R_1R_2R_3$ on the right hand abelian group is defined via conjugation in $F$. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology.

math.GR↗

arXiv2008

Homotopy types of reduced 2-nilpotent simplicial groups

We classify the homotopy types of reduced 2-nilpotent simplicial groups in terms of the homology an d boundary invariants $b,β$. This contains as special cases results of J.H.C. Whitehead on 1-connected 4-dimensional complexes and of Quillen on reduced 2-nilpotent rational simplicial groups. Moreover it yields for 1-nilpotent (or abelian) simplicial groups a classification due to Dold-Kan. Our result describes a new natural structure of the integral homology of any simply connected space. We also classify the homotopy types of connective spectra in the category of 2-nilpotent simplicial groups. Moreover we compute homotopy groups of spheres in the category of $m$-nilpotent groups for $m=2,3$ and partially for $m=4,5$.

math.KT↗

arXiv2007

On certain questions of the free group automorphisms theory

Certain subgroups of the groups $Aut(F_n)$ of automorphisms of a free group $F_n$ are considered. Comparing Alexander polynomials of two poly-free groups $Cb_4^+$ and $P_4$ we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from \cite{CVW}. The questions of linearity of subgroups of $Aut(F_n)$ are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type $Aut(G*\mathbb Z)$ for certain groups $G$ and the subgroup of $IA$-automorphisms $IA(F_n)\subset Aut(F_n)$ are not linear for $n\geq 3$. This generalizes the recent result of Pettet that $IA(F_n)$ are not linear for $n\geq 5$.

math.GR↗