A generalization of the diameter bound of Liebeck and Shalev for finite simple groups

A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an earlier theorem of the authors and Short. We also propose two conjectures that relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in $SL_n(q)$, as well as to the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev.

New Beauville surfaces and finite simple groups

In this paper we construct new Beauville surfaces with group either $\PSL(2,p^e)$, or belonging to some other families of finite simple groups of Lie type of low Lie rank, or an alternating group, or a symmetric group, proving a conjecture of Bauer, Catanese and Grunewald. The proofs rely on probabilistic group theoretical results of Liebeck and Shalev, on classical results of Macbeath and on recent results of Marion.

有限单群中子集乘积的增长

Growth of products of subsets in finite simple groups

本文研究有限非阿贝尔单群中子集与正规子集乘积的增长，证明在两者均不过大时满足 $|AB|\geq|A||B|^{1-\epsilon}$，强化了 Liebeck、Schul、Shalev 的定理。

On the involution fixity of simple groups

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$, which is the maximum number of fixed points of an involution. In this paper we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$. This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), wh…

Bounded normal generation for projective unitary groups of certain infinite operator algebras

We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.

Bounded Normal Generation and Invariant Automatic Continuity

We study the question how quickly products of a fixed conjugacy class in the projective unitary group of a II${}_1$-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the $1$-norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a II${}_1$-factor to a polish SIN group is continuous. Moreover, we show that the projective unitary group of a II${}_1$-factor carries…

On the involution fixity of exceptional groups of Lie type

The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree $n$ is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type. We show that if $T$ is the socle of such a group, then either ${\rm ifix}(T) > n^{1/3}$, or ${\rm ifix}(T) = 1$ and $T = {}^2B_2(q)$ is a Suzuki group in its natural $2$-transitive action of degree $n=q^2+1$. This bound is best possible and we present more detailed results for each family of exceptional groups, which allows us to determine the groups with ${\…

通过超收缩性改进对称群共轭类的覆盖结果

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

本文研究对称群正规子集的覆盖数，通过结合特征标界、Witten zeta函数渐近性与超收缩性定理，将双重指数界限从1/4改进至2/5，并为正规Cayley图中的独立集行为提供了新见解。

具有交错、古典或零散基座的几乎单群的第二极大子群的结构与生成

The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle

针对基座为交错、古典或零散群的几乎单群，修正了其非抛物极大子群生成元数量的分类，将生成元界优化至最优值，确定了相关主因子并改进了已有结论。

简单代数群的拓扑生成

Topological generation of simple algebraic groups

本文研究简单代数群中由共轭类元素生成 Zariski 稠密子群的问题，对辛群和正交群给出完整解，并应用于忠实表示的一般自由性及有限简单群概率生成的新渐近结果。

Beauville surfaces and probabilistic group theory

A Beauville surface is a complex algebraic surface that can be presented as a quotient of a product of two curves by a suitable action of a finite group. Bauer, Catanese and Grunewald have been able to intrinsically characterize the groups appearing in minimal presentations of Beauville surfaces in terms of the existence of a so-called "Beauville structure". They conjectured that all finite simple groups, except A5, admit such a structure. This conjecture has recently been proved by Guralnick-Malle and Fairbairn-Magaard-Parker. In this survey we demonstrate another approach towards the proof o…

对称群的强超压缩性及其应用

Sharp hypercontractivity for symmetric groups and its applications

本文研究对称群和交错群Cayley图的组合性质，通过将正规条件推广为全局伪随机条件，并利用全局函数卷积算子的谱估计，结合新的强超压缩不等式，推广了多项经典结论并给出了直径问题的本质最优估计。