$(2,p)$-生成群的图正则表示

Cubic Graphical Regular Representations of $\mathrm{PSU}_3(q)$

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. Towards a proof of the conjecture that only finitely many finite simple groups have no cubic GRR, this paper shows that $\mathrm{PSU}_3(q)$ has a cubic GRR if and only if $q\geq4$. Moreover, a cubic GRR of $\mathrm{PSU}_3(q)$ is constructed for each of these $q$.

图形正则表示的渐近计数

Asymptotic enumeration of graphical regular representations

本文估计了阶足够大的群的图形正则表示数量，证明了几乎所有有限Cayley图的自同构群都尽可能小，从而证实了Babai-Godsil-Imrich-Lovasz和Xu关于GRRs及正规Cayley图比例的两个猜想。

Cubic graphical regular representations of $PSL_2(q)$

We study cubic graphical regular representations of the finite simple groups $PSL_2(q)$. It is shown that such graphical regular representations exist if and only if $q\neq7$, and the generating set must consist of three involutions.

图小消去理论中的优化

Optimization in graphical small cancellation theory

该研究聚焦图小消去理论，简化Osajda的概率方法，在构造含扩张图族的Gromov怪物群时显著减少生成元数量，推进相关群构造的优化。

Cubic graphical regular representations of some classical simple groups

A graphical regular representation (GRR) of a group $G$ is a Cayley graph of $G$ whose full automorphism group is equal to the right regular permutation representation of $G$. In this paper we study cubic GRRs of $\mathrm{PSL}_{n}(q)$ ($n=4, 6, 8$), $\mathrm{PSp}_{n}(q)$ ($n=6, 8$), $\mathrm{P}Ω_{n}^{+}(q)$ ($n=8, 10, 12$) and $\mathrm{P}Ω_{n}^{-}(q)$ ($n=8, 10, 12$), where $q = 2^f$ with $f \ge 1$. We prove that for each of these groups, with probability tending to $1$ as $q \rightarrow \infty$, any element $x$ of odd prime order dividing $2^{ef}-1$ but not $2^{i}-1$ for each $1 \le i < ef$ t…

图离散群与刚性

Graphically discrete groups and rigidity

本文引入群的图离散性概念，证明顶点群为图离散群的群图具强刚性，包括自由积的作用刚性与特定双曲群图的拟等距刚性，还给出例子并指出其非可公度不变性。