Related papers: On regular subgroups of $\mathsf{SL}_3(\mathbb{R})…

Let p be an odd prime. The lattice of all normal subgroups and the terms of the lower and upper central series are determined for all metabelian p-groups with generator rank d=2 having abelianization of type (p,p) and minimal defect of…

We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over an Artinian ring containing the group of…

We determine the normalizer in $SL_{2}(\mathbb{R})$ of several families of congruence subgroups of $SL_{2}(\mathbb{Z})$. In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels…

We consider three families of groups: the Bianchi groups SL(2,O) where O is the ring of integers of an imaginary, quadratic field; the groups SL*(2,O) where O is a *-order of a definite, rational quaternion algebra with an orthogonal…

Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the…

For a finite group $G$, let $\sigma(G)$ be the number of subgroups of $G$ and $\sigma_\iota(G)$ the number of isomorphism types of subgroups of $G$. Let $L=L_r(p^e)$ denote a simple group of Lie type, rank $r$, over a field of order $p^e$…

Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously…

This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski,…

Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact set. As a…

We investigate the question of how many subgroups of a finite group are not in its Chermak-Delgado lattice. The Chermak-Delgado lattice for a finite group is a self-dual lattice of subgroups with many intriguing properties. Fasol\u{a} and…

We construct Galois theory for sublattices of certain complete modular lattices and their automorphism groups. A well-known description of the intermediate subgroups of the general linear group over a semilocal ring containing the group of…

We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the $l^1$-homology of a word-hyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a…

Let $\Gamma$ be a countable group and $\mathrm{Sub}(\Gamma)$ its Chabauty space, namely the compact $\Gamma$-space consisting of all subgroups of $\Gamma$. We call a subgroup $\Delta \in \mathrm{Sub}(\Gamma)$ a boomerang subgroup if for…

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…

Let $\Gamma$ be a non-elementary Kleinian group and $H<\Gamma$ a finitely generated, proper subgroup. We prove that if $\Gamma$ has finite co-volume, then the profinite completions of $H$ and $\Gamma$ are not isomorphic. If $H$ has finite…

We initiate the study of p-adic algebraic groups G from the stability-theoretic and definable topological-dynamical points of view, that is, we consider invariants of the action of G on its space of types over Q_p in the language of fields.…

We construct certain subgroups of hyperbolic triangle groups which we call "congruence" subgroups. These groups include the classical congruence subgroups of SL_2(ZZ), Hecke triangle groups, and 19 families of arithmetic triangle groups…

Let $\overline{M}$ be a compact smoothly stratified pseudo-manifold endowed with a wedge metric $g$. Let $\overline{M}_\Gamma$ be a Galois $\Gamma$-covering. Under additional assumptions on $\overline{M}$, satisfied for example by Witt…

We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…