Profinite and Finite Groups Associated with Loop and Diffeomorphism Groups of Non-Archimedean Manifolds

A structure and representations of diffeomorphism groups of non-Archimedean manifolds

Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces over non-Archimedean fields $\bf F$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex manifolds. It is proved that $G$ is not a Banach-Lie group, but it has a neighbourhood $W$ of the unit element $e$ such that each element $g$ in $W$ belongs to at least one corresponding one-parameter subgroup. It is proved that $G$ is simple and perfect. Its compact subgroups $G_c$ are studied such that a dimension over $\bf F$ of its tangent space $dim_{\bf F}T…

On profinite subgroups of an algebraic group over a local field

The purpose of this paper is to link anisotropy properties of an algebraic group together with compactness issues in the topological group of its rational points. We nd equivalent conditions on a smooth ane algebraic group scheme over a non-Archimedean local eld for the associated rational points to admit maximal compact subgroups. We use the structure theory of pseudo-reductive groups provided, whatever the characteristic, by Conrad, Gabber and Prasad. We also investigate thoroughly maximal prop subgroups in the semisimple case, using Bruhat-Tits theory.

Protori 与无挠阿贝尔群

Protori and Torsion-Free Abelian Groups

本文利用紧致阿贝尔群分解定理证明 protorus 的 profinite 子群构成格，引入非阿基米德维数作为分类不变量，并研究其与无挠阿贝尔群的联系。

Free non-archimedean topological groups

We study free topological groups defined over uniform spaces in some subclasses of the class NA of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean NA groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another contrasting advantage is that the induced topological group actions on free abelian NA group…

On the procongruence completion of the Teichmüller modular group

For $2g-2+n>0$, the Teichmüller modular group $Γ_{g,n}$ of a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$ is the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let $Π_{g,n}$ be the fundamental group of $S_{g,n}$, with a given base point, and $\hatΠ_{g,n}$ its profinite completion. There is then a natural faithful representation $Γ_{g,n}\hookrightarrow Out(\hatΠ_{g,n})$. The procongruence completion $\checkΓ_{g,n}$ of the Teichmüller group is defined to be the closure of the Teic…

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let Γbe an arithmetic lattice in G and let C=C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \ha…