Related papers: Free subgroups of special linear groups

A LAnKe (also known as a Lie algebra of the $n$th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric $n$-linear form that satisfies the generalized Jacobi identity. The symmetric group $\mathfrak{S}_m$ acts on the…

In algebraic number theory, the finiteness of the Picard group of an order in a number field is generally proved via a lattice argument: the order forms a lattice and every ideal class contains an integral ideal with a small enough non-zero…

We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups. Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a…

Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic…

Let G be any locally compact, unimodular, metrizable group. The main result of this paper, roughly stated, is that if F<G is any finitely generated free group and \Gamma < G any lattice, then up to a small perturbation and passing to a…

In this paper we find a characterization for groups elementarily equivalent to a free nilpotent group $G$ of class 2 and arbitrary finite rank.

For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.

In this article we relate two different densities. Let $F_k$ be the free group of finite rank $k \ge 2$ and let $\alpha$ be the abelianization map from $F_k$ onto $ \mathbb{Z}^k$. We prove that if $S \subseteq \mathbb{Z}^k$ is invariant…

We describe all finite subsemigroups of a free left regular band of infinite rank. Moreover, we show applications of this result in algebraic geometry and model theory.

Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, \ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i \in C_i$ such that…

We classify all possible JSJ decompositions of doubles of free groups of rank two and we then compute the Makanin-Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup…

We prove that every verbally closed subgroup of a free group $F$ of a finite rank is a retract of $F.$

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups $G$ has infinite palindromic width, provided that $G$ is not the free product of two cyclic groups…

We obtain an infinite family of orthogonal hypergeometric groups, which are higher rank arithmetic groups. We also list cases of arithmetic hypergeometric groups whose real Zariski closure is O(2,3).

A group $G$ is said to have dense ${\cal CD}$-subgroups if each non-empty open interval of the subgroup lattice $L(G)$ contains a subgroup in the Chermak--Delgado lattice ${\cal CD}(G)$. In this note, we study finite groups satisfying this…

In this paper we study decidability and complexity of decision problems on matrices from the special linear group $\mathrm{SL}(2,\mathbb{Z})$. In particular, we study the freeness problem: given a finite set of matrices $G$ generating a…

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of $GL(n, K)$ ($K$ - a field, $n \geq 3$) which is not contained in the center, contains $SL(n, K)$. A. Rosenberg gave description of normal subgroups…

Suppose $\ell$ is a prime number, $\ell >3$, $K$ is a field that is an unramified finite extension of the field $\Q_\ell$ of $\ell$-adic numbers, and $G$ is a finite group that is a semi-direct product of a normal $\ell'$-subgroup $H$ and a…

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…

For an element $g$ of a group $G$, a right Engel sink of $g$ is a subset of $G$ containing all sufficiently long commutators $[...[[g ,x],x],\dots ,x]$ for all $x\in G$. A left Engel sink of $g$ is a subset of $G$ containing all…