$p$ -自变数上的多项式映射与群自同态映射环的剩余性质

自由群自同构的映射环面是连贯的

Mapping tori of free group automorphisms are coherent

群G的一个自同态Φ的映射环面是HNN-扩张G*_G，其键映射是单位映射和Φ。证明了内射自由群自同态的映射环面具有其生成子群是双表示的性质，并且这些子群是有限型的。

On self-similarity of $p$-adic analytic pro-$p$ groups of small dimension

Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular rooted trees can be studied through the virtual endomorphisms of the associated $\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable $\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$, and to determine which of them admit a faithful self-similar actio…

p-adic对称空间的极式分解

Polar decomposition for p-adic symmetric spaces

设G是连通可约k-群的k-点群，H是G的对合s的对称子群.本文证明了特征不为2的任意局部域k上的对称空间G/H的一个极式分解G = KAH。这里K是G的紧子集，A是极大（k，s）-分裂环面的k-点群的有限并，每个H-共轭类有一个。这种分解类似于实对称空间G/H的著名的极式分解G = KAH。

Rational Conjugacy Classes of Maximal Tori In Groups of Type $D_n$

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type $D_n$, with specific emphasis on the case of number fields and p-adic fields. This includes the forms associated to quadratic spaces, all of their inner and outer forms as well as the Spin groups, their simply connected covers. In particular, in this work we handle all (simply connected) outer forms of $D_4$.

Brace blocks from bilinear maps and liftings of endomorphisms

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families of group operations on a set $G$ such that any two of them induce a skew brace structure on $G$. We construct these operations by using bilinear maps and liftings of endomorphisms of quotient groups with respect to a central subgroup. We provide several examples of the construction, showing that there are brace blocks which consist of distinct operations of any given cardinality. One of the examples we give yields an answer to a question of Cornelius Greither. This example exhibits a seque…

A look at the inner structure of the $2$-adic ring $C^*$-algebra and its automorphism groups

We undertake a systematic study of the so-called $2$-adic ring $C^*$-algebra $\mathcal{Q}_2$. This is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_2$ such that $S_2U=U^2S_2$ and $S_2S_2^*+US_2S_2^*U^*=1$. Notably, it contains a copy of the Cuntz algebra $\mathcal{O}_2=C^*(S_1, S_2)$ through the injective homomorphism mapping $S_1$ to $US_2$. Among the main results, the relative commutant $C^*(S_2)'\cap \mathcal{Q}_2$ is shown to be trivial. This in turn leads to a rigidity property enjoyed by the inclusion $\mathcal{O}_2\subset\mathcal{Q}_2$, namely the endomorphis…

经典$p$-进李群与李代数的自相似性

Self-similarity of the classical $p$-adic Lie groups and Lie algebras

该研究通过分析经典李格的虚自同态，结合正交群主同余子群指数的计算，得到无穷多组分歧指数，证明p-进域整数环上的经典李群可在对应正则有根叉树上实现第一层传递的忠实自相似作用。

Normalisers of maximal tori and a conjecture of Vdovin

Let $G = O^{p'}(\bar{G}^F)$ be a finite simple group of Lie type defined over a field of characteristic $p$, where $F$ is a Steinberg endomorphism of the ambient simple algebraic group $\bar{G}$. Let $\bar{T}$ be an $F$-stable maximal torus of $\bar{G}$ and set $N = N_G(\bar{T})$. A conjecture due to Vdovin asserts that if $G \not\cong {\rm L}_3(2)$ then $N \cap N^x$ is a $p$-group for some $x \in G$. In this paper, we use a combination of probabilistic and computational methods to calculate the base size for the natural action of $G$ on $G/N$, which allows us to prove a stronger, and suitably…

G2的某些子群的分类

Classification of Certain Subgroups of G2

我们给出了G2型群中极大环面的有理共轭类的一个具体刻画，重点讨论了数域和p-adic域的情形。在同样的上下文中，我们刻画了G2的A2子群的有理共轭类。在得到了具体的特征之后，我们将其与可以用伽罗瓦上同调给出的更抽象的特征联系起来。我们注意到，这些结果对A2子群同时和独立开发的工作Hooda而结果对环面同时和独立开发的工作贝利-Gille-李。

p-adic约化对称空间中Cartan分解的一个类似

An analogue of the Cartan decomposition for p-adic reductive symmetric spaces

设F是一个非阿基米德局部紧域，其剩余特征不同于2，G是定义在F上的一个连通约化群，s是G的一个对合F-自同构，H是s的不动点群的开F-子群.我们记为G（F）（resp. H（F））是G的F-点群（分别为H）。本文得到了约化对称空间G（F）/H（F）的Cartan分解的一个类似.更精确地说，我们得到了G（F）作为H（F）-陪集的并的一个分解，它与G中极大s-反不变F-分裂环面的H（F）-共轭类有关。当G是F-分裂时，我们得到了一个更精确的结果，涉及到G在F上的Bruhat-Tits建筑的一个特殊点的稳定子。

Solenoid Maps, Automatic Sequences, Van Der Put Series, and Mealy-Moore Automata

The ring $\mathbb Z_d$ of $d$-adic integers has a natural interpretation as the boundary of a rooted $d$-ary tree $T_d$. Endomorphisms of this tree (i.e. solenoid maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_d$ to itself and automorphisms of $T_d$ constitute the group $\mathrm{Isom}(\mathbb Z_d)$. In the case when $d=p$ is prime, Anashin showed that $f\in\mathrm{Lip}^1(\mathbb Z_p)$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a $p$-automatic sequence over a finite subset of $\mathbb Z_p\…