Torsion Free Endotrivial Modules for Finite Groups of Lie Type

Relatively free nilpotent torsion-free groups and their Lie algebras

For a torsion free finitely generated nilpotent group G we naturally associate four finite dimensional nilpotent Lie algebras over a field of characteristic zero. We show that if G is a relatively free group of some variery of nilpotent groups then all the above Lie algebras are isomorphic. As a result, any two quasi-isometric relatively free nilpotent groups are isomorphic. Moreover let L be a relatively free nilpotent Lie algebra over Q generated by X. We give L the structure of a group by means of the Baker-Campbell-Hausdorff formula and we show that the subgroup H generated by X is relativ…

Torsion subgroups, solvability and the Engel condition in associative rings

The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.

Generic properties of 2-step nilpotent Lie algebras and torsion-free groups

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained in a proper Zariski-closed subset. We compute the maximal dimension of an abelian subalgebra of a generic Lie algebra and give a sufficient condition for a generic Lie algebra to admit no surjective homomorphism onto a non-abelian Lie algebra of a given dimension. Also we consider analogous questions for finitely generated torsion free nilpotent groups of cl…

Lie algebras and Higher torsion in p-groups

We study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence E_r^{*,*}[g] is defined for any Lie algebra g which models the Bockstein spectral sequence of the corresponding group in characteristic p. This spectral sequence is then studied for complex semisimple Lie algebras like sl_n(C), and the results there are transferred to the corresponding p-group via the intermediary arithmetic Lie algebra defined over Z. Over C, it is shown that E_1^{*,*}[g]=H^*(g,U(g)^*)=H^*(ΛBG) where U(g)^* is the dual of the universal envelop…

L2-invariants of nonuniform lattices in semisimple Lie groups

We compute L2-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov-Shubin numbers and vanishing L2-torsion for lattices in groups with even deficiency. We discuss applications to Gromov's Zero-in-the-Spectrum Conjecture as well as to a proportionality conjecture for the L2-torsion of measure equivalent groups.

Approximate groups, I: the torsion-free nilpotent case

We describe the structure of ``K-approximate subgroups'' of torsion-free nilpotent groups, paying particular attention to Lie groups. Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which independently address related issues. We comment briefly on some of the connections between these papers.

Gradings for nilpotent Lie algebras

We present a constructive approach to torsion-free gradings of Lie algebras. Our main result is the computation of a maximal grading. Given a Lie algebra, using its maximal grading we enumerate all of its torsion-free gradings as well as its positive gradings. As applications, we classify gradings in low dimension, we consider the enumeration of Heintze groups, and we give methods to find bounds for non-vanishing $\ell^{q,p}$ cohomology.

Group and Lie algebra filtrations and homotopy groups of spheres

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the "dimension problem" by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary $s,d$ the torsion of the homotopy group $π_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, since for every prime $p$, there is some $p$-torsion in $π_{2p}(S^2)$ by a result of Serre. We explai…

Analytic pro-p groups of small dimensions

According to Lazard, every p-adic Lie group contains an open pro-p subgroup which is saturable. This can be regarded as the starting point of p-adic Lie theory, as one can naturally associate to every saturable pro-p group G a Lie lattice L(G) over the p-adic integers. Essential features of saturable pro-p groups include that they are torsion-free and p-adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p-adic analytic pro-p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance,…

推广Lichtman定理

Generalising a Theorem of Lichtman

本文在适当附加条件下，证明了自由合并积的限制Zassenhaus F_p-李代数或有理Magnus李代数是各因子对应李代数的自由合并积，推广了Lichtman关于自由积的定理，并应用于剩余性质与上同调完全性问题。

On torsion in finitely presented groups

We give a uniform construction that, on input of a recursive presentation $P$ of a group, outputs a recursive presentation of a torsion-free group, isomorphic to $P$ whenever $P$ is itself torsion-free. We use this to re-obtain a known result, the existence of a universal finitely presented torsion-free group; one into which all finitely presented torsion-free groups embed. We apply our techniques to show that recognising embeddability of finitely presented groups is $Π^{0}_{2}$-hard, $Σ^{0}_{2}$-hard, and lies in $Σ^{0}_{3}$. We also show that the sets of orders of torsion elements of finitel…

Rank, combinatorial cost and homology torsion growth in higher rank lattices

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute. Most non-uniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (co-co…

用马格努斯李代数表示扎森豪斯李代数

Presenting the Zassenhaus Lie algebra by the Magnus Lie algebra

本文证明（pro-p）群的扎森豪斯限制$\mathbb F_p$-李代数可由其马格努斯李代数表示，对下中心列项无挠的群类可显式计算，结果适用于曲面群等多类群。

Contractible Lie groups over local fields

Let G be a Lie group over a local field of positive characteristic which admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each group element x converge to the neutral element 1). We show that then G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results even extend to Lie groups over arbitrary complete ultrametric fields.

Limit theorems for some long range random walks on torsion free nilpotent groups

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $Γ$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $μ$, we construct a homogeneous nilpotent Lie group $G_\bullet(Γ,μ)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on th…

Hamiltonian no-torsion

In 2002 Polterovich has notably established that on closed aspherical symplectic manifolds, Hamiltonian diffeomorphisms of finite order, which we call Hamiltonian torsion, must in fact be trivial. In this paper we prove the first higher-dimensional Hamiltonian no-torsion theorems beyond the symplectically aspherical case. We start by showing that closed symplectic Calabi-Yau and negative monotone symplectic manifolds do not admit Hamiltonian torsion. Going still beyond topological constraints, we prove that every closed positive monotone symplectic manifold $(M,ω)$ admitting Hamiltonian torsio…