Related papers: Index-stable compact $p$-adic analytic groups

A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a…

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$.…

The rank rk(G) of a profinite group G is the supremum of d(H), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure…

We show that for an arbitrary stable theory T, a group G is profinite if and only if G occurs as a Galois group of some Galois extension inside a monster model of T. We prove that any PAC substructure of the monster model of T has…

In this paper we describe some properties of groups $G$ that contain a solvable subgroup of finite prime-power index (Theorem 1 and Corollaries 2--3). We prove that if $G$ is a non-solvable group that contains a solvable subgroup of index…

For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group $\text{Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for…

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if…

Let $K$ be a complete non-trivially valued non-Archimedean field. Given an algebraic group over $K$ on which every regular function is constant, any rigid analytic function is shown to be constant too. It follows that an algebraic group…

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either…

A group $G$ is said to be a $C$-group if every subgroup $H$ has a permutable complement, i.e. if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H \cap K=1$. In this paper, we study the profinite counterpart of this concept. We say…

By arithmeticity and superrigidity, a commensurability class of lattices in a higher rank Lie group is defined by a unique algebraic group over a unique number subfield of $\mathbb{R}$ or $\mathbb{C}$. We prove an adelic version of…

Suppose that $G$ is a finite $p$-group. If all subgroups of index $p^t$ of $G$ are abelian and at least one subgroup of index $p^{t-1}$ of $G$ is not abelian, then $G$ is called an $\mathcal{A}_t$-group. In this paper, some information…

Let $G$ be a group. The prime index graph of $G$, denoted by $\Pi(G)$, is the graph whose vertex set is the set of all subgroups of $G$ and two distinct comparable vertices $H$ and $K$ are adjacent if and only if the index of $H$ in $K$ or…

A subgroup H of a group G is called inert if for each $g\in G$ the index of $H\cap H^g$ in $H$ is finite. We give a classification of soluble-by-finite groups $G$ in which subnormal subgroups are inert in the cases where $G$ has no…

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly…

Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that…

Let $\mathbb{K}$ be an algebraically closed field of characteristic 0. A finite dimensional Lie algebra $\mathfrak{g}$ over $\mathbb{K}$ is said to be stable if there exists a linear form $g\in\mathfrak{g}^{*}$ and a Zariski open subset in…

We show that if $G$ is a sufficiently saturated stable group of finite weight with no infinite, infinite-index, chains of definable subgroups, then $G$ is superstable of finite $U$-rank. Combined with recent work of Palacin and Sklinos, we…

A locally compact group $G$ is said to be $\ast$-regular if the natural map $\Psi:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$ is a homeomorphism with respect to the Jacobson topologies on the primitive ideal spaces $\Prim C^\ast(G)$ and…

A group may be considered $C^*$-stable if almost representations of the group in a $C^*$-algebra are always close to actual representations. We initiate a systematic study of which discrete groups are $C^*$-stable or only stable with…