Related papers: HR-length of a free group via polynomial functors

In this paper, the $H\mathbb Z$-length of different groups is studied. By definition, this is the length of $H\mathbb Z$-localization tower or the length of transfinite lower central series of $H\mathbb Z$-localization. It is proved that,…

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in…

We give a criterion for bounding the homological finiteness length of certain HF-groups. This is used in two distinct contexts. Firstly, the homological finiteness length of a non-uniform lattice on a locally finite n-dimensional…

We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places…

Let $h$ be a connective homology theory. We construct a functorial relative plus construction as a Bousfield localization functor in the category of maps of spaces. It allows us to associate to a pair $(X, H)$ consisting of a connected…

The homological properties of localizations and completions of metabelian groups are studied. It is shown that, for $R=\mathbb Q$ or $R=\mathbb Z/n$ and a finitely presented metabelian group $G$, the natural map from $G$ to its…

We strengthen Marshall Hall's Theorem to show that free groups are locally extended residually alternating. Let F be any free group of rank at least two, let H be a finitely generated subgroup of infinite index in F and let {g_1,...,g_n} be…

We show that it is relatively consistent with ZF that the Borel hierarchy on the reals has length $\omega_2$. This implies that $\omega_1$ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument…

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in…

Let $G$ be a finite soluble group and $h(G)$ its Fitting length. The aim of this paper is to give certain upper bounds for $h(G)$ as functions of the Fitting length of at least three Hall subgroups of $G$ which factorize $G$ in a particular…

Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of dimension at least $2$ such that the isometry group of $X$ has property $(T)$. We prove that for every torsion free lattice $\Gamma\subset {\rm Isom} X$ any homology…

Let $\mathsf{F}_r$ be a free group of rank $r$, $\mathbb{F}_q$ a finite field of order q, and let $\mathrm{SL}_n(\mathbb{F}_q)$ act on $\mathrm{Hom}(\mathsf{F}_r, \mathrm{SL}_n(\mathbb{F}_q))$ by conjugation. We describe a general algorithm…

In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are…

We show that the free group of rank 2 is a limit of 2-markings of Thompson's group F in the space of all 2-marked groups. More specifically, we find a sequence of generating pairs for F so that as one goes out the sequence, the length of…

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: \Omega \text {Ham}(M, \omega) \to \mathbb{R}$, with Gromov-Witten/Floer…

For any finitely generated group $G$, two complexity functions $\alpha_G$ and $\beta_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its…

Let G(O_S) be an S-arithmetic subgroup of a connected, absolutely almost simple linear algebraic group G over a global function field K. We show that the sum of local ranks of G determines the homological finiteness properties of G(O_S)…

We prove that the length of the shortest identity in a finite simple group of Lie type of rank $r$ defined over $\mathbb{F}_q$, is bounded (from above and below) by explicit polynomials in $q$ and $r$.

For a group $G$ and $R=\mathbb Z,\mathbb Z/p,\mathbb Q$ we denote by $\hat G_R$ the $R$-completion of $G.$ We study the map $H_n(G,K)\to H_n(\hat G_R,K),$ where $(R,K)=(\mathbb Z,\mathbb Z/p),(\mathbb Z/p,\mathbb Z/p),(\mathbb Q,\mathbb…

A theorem by Hall asserts that the multiplication in torsion free nilpotent groups of finite Hirsch length can be facilitated by polynomials. In this note we exhibit explicit Hall polynomials for the torsion free nilpotent groups of Hirsch…