Related papers: Topological generation of exceptional algebraic gr…

We prove group existence and structure theorems in a general setting of tame topological theories. More precisely, we identify a linear/non-linear dividing line -- called topological 1-basedness -- among the class of t-minimal theories with…

Throughout this Abstract, $G$ is a topological Abelian group and $\hat{G}$ is the space of continuous homomorphisms from $G$ into $T$ in the compact-open topology. A dense subgroup $D$ of $G$ determines $G$ if the (necessarily continuous)…

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group $G$ which contains a higher rank lattice is a lattice in the group $G$. For example, we show that a Zariski dense subgroup of…

Let $G$ be a connected, absolutely almost simple, algebraic group defined over a finitely generated, infinite field $K$, and let $\Gamma$ be a Zariski dense subgroup of $G(K)$. We show, apart from some few exceptions, that the…

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show…

In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\dots,g_k$ with order…

Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its…

Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.

We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the…

We study lattices in a product $G = G_1 \times \dots \times G_n$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_i$ is…

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of…

A subset S of a finite group G invariably generates G if G = <hsg(s) j s 2 Si > for each choice of g(s) 2 G; s 2 S. We give a tight upper bound on the minimal size of an invariable generating set for an arbitrary finite group G. In response…

Famously, every finite simple group $G$ can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate $G$ tends to $1$ as $|G| \to \infty$. More generally, work of…

A topological group is locally pseudocompact if it contains a non-empty open set with pseudocompact closure. In this note, we prove that if G is a group with the property that every closed subgroup of G is locally pseudocompact, then G_0 is…

Let G be any abelian group and {a_sG_s}_{s=1}^k be a finite system of cosets of subgroups G_1,...,G_k. We show that if {a_sG_s}_{s=1}^k covers all the elements of G at least m times with the coset a_tG_t irredundant then [G:G_t]\le 2^{k-m}…

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the…

Let $G$ be an acylindrically hyperbolic group. We prove that if $G$ has no non-trivial finite normal subgroups, then the set of invertible elements is dense in the reduced $C^\ast$-algebra of $G$. The same result is obtained for finite…

Exotic group $C^*$-algebras are $C^*$-algebras that lie between the universal and the reduced group $C^*$-algebra of a locally compact group. We consider simple Lie groups $G$ with real rank one and investigate their exotic group…

We describe an algorithm for determining the algebraic subgroup of GL(n,C) that is defined as the closure of the group generated by a finite number of elements of GL(n,C). The algorithm avoids the use of Groebner bases and can be used on…

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of…