Related papers: Orbits of Free Cyclic Submodules over Rings of Low…

The *somewhere-to-below shuffles* are the elements \[ t_{\ell} := \operatorname{cyc}_{\ell}+\operatorname{cyc}_{\ell,\ell+1}+\operatorname{cyc}_{\ell,\ell+1,\ell+2}+\cdots+\operatorname{cyc}_{\ell,\ell+1,\ldots,n} \] (for $\ell \in…

We consider the action of a parabolic subgroup of the General Linear Group on a metabelian ideal. For those actions, we classify actions with finitely many orbits using methods from representation theory.

We give analogues in the finite general linear group of two elementary results concerning long cycles and transpositions in the symmetric group: first, that the long cycles are precisely the elements whose minimum-length factorizations into…

We prove that if every Schmidt subgroup of a group $G$ is subnormal or modular, then $G/F(G)$ is cyclic

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete…

A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type…

We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on C^n using the extended limit sets (the J-sets). Moreover we construct for any n\geq 2 an abelian semigroup G of GL(n;C) generated by n + 1…

It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of…

We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.

We construct a new family of simple $\mathfrak{gl}_{2n}$-modules which depends on $n^2$ generic parameters. Each such module is isomorphic to the regular $U(\mathfrak{gl}_{n})$-module when restricted the $\mathfrak{gl}_{n}$-subalgebra…

We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation…

We take a direct approach to computing the orbits for the action of the automorphism group $\mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{\mathbb{G}_2} \cong…

For square-free positive integers $n$, we study the action of the modular group $\mbox{PSL}(2,\mathbb{Z})$ on the subsets $\{\,\frac{a+\sqrt{-n}}{c}\in \mathbb{Q}(\sqrt{-n})\, | \, a,b=\frac{a^2+n}{c},c \in \mathbb{Z} \,\}$ of the imaginary…

Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is…

Let $G$ be a group acting freely, properly discontinuously and cellularly on a finite dimensional $C$W-complex $\Sigma(2n)$ which has the homotopy type of the $2n$- sphere $\mathbb{S}^{2n}$. Then, this action induces an action of the group…

Let R be an n-dimensional Cohen-Macaulay local ring and Q a parameter ideal of R. Suppose that an acyclic complex (F_{\bullet}, \varphi_{\bullet}) of length n of finitely generated free R-modules is given. We put M = Im \varphi_{1}, which…

Let $G$ be a finite group of odd order, $\F$ a finite field of odd characteristic $p$ and $\B$ a finite--dimensional symplectic $\F G$-module. We show that $\B$ is $\F G$-hyperbolic, i.e., it contains a self--perpendicular $\F G$-submodule,…

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…

For the cyclic group $C_2$ we give a complete description of the derived category of perfect complexes of modules over the constant Mackey ring $\underline{\mathbb{Z}/\ell}$, for $\ell$ a prime. This is fairly simple for $\ell$ odd, but for…

Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose…