所有非幂零子群均可置换的局部分次群，II

关于接纳一个值为恩格尔的词的群

On groups admitting a word whose values are Engel

设m，n是正整数，v是多线性换位子字，w = v^m.证明了：如果G是剩余有限群，且其中所有w-值都是n-Engel，则动词子群w（G）是局部幂零的.我们还研究的问题，这是否是真的情况下，G是局部分级，而不是剩余有限的。在m = 1的情况下，我们肯定地回答这个问题。进一步证明了如果u是非换位子字，G是局部分次群且u-值都是n-Engel，则动词子群u（G）是局部幂零的.

Cohomological induction and uniform measure equivalence

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric, connected, simply connected, nilpotent Lie groups are isomorphic. This unifies results of Shalom and Sauer and also provides new insight into the quasi-isometry classification problem for low dimensional nilpotent Lie groups.

Sylow 数与有限群的结构

Sylow numbers and the structure of finite groups

本文研究有限群 $G=AB$ 为两个子群互置换乘积的情形，利用 $A$ 和 $B$ 的 Sylow 数给出 $p$-长度、幂零长度和导长度的新界，推广了 Zhang 的已知结果。

Compatible filters with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

有限群的σ-性质的多项式时间判定

$σ$-properties of finite groups in polynomial time

本文证明了对置换群商群G/K的σ-幂零性、σ-可解性及子群商H/K的σ-次正规性、σ-p-可置换性均可在多项式时间内判定，并能求出相应的最小划分σ。

Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^m$, $q^{-s}$, and $q^{-ms}$ giving the ideal zeta function of the amalgamation of $m$ copies of $L$ over the derived subring, for every $m \geq 1$, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver rep…

关于幂零群的遍历族的渐近稳定性

Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

设Gamma是一个n-生成的幂零群.我们考虑三个密切相关的问题：（i）Gamma上等变遍历族的渐近锥面，推广了Pansu定理;（ii）Gamma的Cayley图的边上一般遍历过程（不一定独立）的首次通过渗流的极限形状;（iii）一般遍历Gamma作用的次可加遍历定理。极限对象是在与Gamma的Mal'cev完备化相关的分次幂零群群上的Carnot-Caratheodory度量中给出的.

具有剩余性质的群的诣零自同构

Nil-automorphisms of groups with residual properties

根据Plotkin，我们说群$G$的自同构$x$是一个零自同构，如果对于每个$g\in G$，存在$n=n(g)$使得$[g,_n x]=1$。如果整数$n$可以独立于$g$来选择，则称$x$是幂单的。零自同构和幂单自同构可以看作Engel元概念的自然扩张，因为零自同构$x$就是$G < x >$中的左Engel元.本文研究了具有剩余性质的群即局部分次群、剩余有限群和投射有限群的诣零自同构。我们证明的第一个结果是：局部分次群的诣零自同构的有限群必为幂零群。接下来，我们把注意力转向剩余有限群和投射有限群的幂单自同构群。我们表明，这样的群体是局部幂零的，作为一个副产品，我们获得了一个著名的威尔逊定理的替代证明关于n-恩格尔剩余有限群。

On residually finite groups satisfying an Engel type identity

Let $ n, q $ be positive integers. We show that if $ G $ is a finitely generated residually finite group satisfying the identity $ [x,_ny^q]\equiv 1, $ then there exists a function $ f(n) $ such that $ G $ has a nilpotent subgroup of finite index of class at most $ f(n) $. We also extend this result to locally graded groups.

关于幂零群中心积的Dehn函数

On the Dehn functions of central products of nilpotent groups

本文确定了多类幂零李群中心积的Dehn函数，证实了相关猜想，并构造了无限族群使Cornulier的界渐近最优。

幂零群的$L^p$测度等价

$L^p$ measure equivalence of nilpotent groups

本文对所有$p\leq 1$分类了多项式增长局部紧群的$L^p$测度等价，构造了单连通幂零李群间的显式轨道等价，并首次给出$p>1$时的刚性结果，推广了相关定理。

Formality properties of finitely generated groups and Lie algebras

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the afor…

幂零群拟等距的上同调与乘法间隙

Cohomology and multiplicative gaps for quasi-isometries of nilpotent groups

该研究在七维无挠幂零格上构造两个等距渐近锥的字度量，利用拟等距的凸不变量与上同调相容性，证明其拟等距映射存在至少1+ε的乘法间隙。