On the self-similarity index of $p$-adic analytic pro-$p$ groups

p-adic群的自相似性

Self-similarity of p-adic groups

本文证明了单代数p-adic群的紧开子群的自相似性等价于其迷向性。

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。

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…

Similarity classes of integral $p$-adic matrices and representation zeta functions of groups of type $A_2$

We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta functions of arithmetic groups $Γ\subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form of $\mathsf{SL}_3$: assuming that $Γ$ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of $Γ$. Our results are based on explicit, uniform formulae for the representati…

On the self-similarity of the norm one group of $p$-adic division algebras

Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of $SL_1(D)$ admits self-similar actions on regular rooted trees. Moreover, we prove results on $\mathbb{Z}_p$-Lie lattices that allow to deal with the case where the center of $D$ is bigger than $\mathbb{Q}_p$, and lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial c…

p-adic functionals on torsion-free abelian groups

Let p be a prime. A p-adic functional on a torsion-free abelian group G is a group homomorphism from G to the p-adic integers. The group of all such p-adic functionals is viewed as a p-adic dual group of G, and is studied from the point of view of functional analysis. An analogue of the Hahn--Banach Theorem is proved; this result shows that there are sufficiently many p-adic functionals to be interesting. There is a natural homomorphism from G to its p-adic double dual, and one main result that is proved is that the image of G in this double dual is dense in an appropriate topology. This is us…