Actions Of Solvable Groups Research Articles

Higher rank confining subsets and hyperbolic actions of solvable groups

Recent papers of the authors have completely described the hyperbolic actions of several families of classically studied solvable groups. A key tool for these investigations is the machinery of confining subsets of Caprace, Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups with virtually cyclic abelianizations. In this paper, we extend this machinery and give a correspondence between the hyperbolic actions of certain solvable groups with higher rank abelianizations and confining subsets of these more general groups. We then apply this extension to give a complete description of the hyperbolic actions of generalized solvable Baumslag-Solitar groups and to reprove a result of Sgobbi-Wong computing their Bieri-Neumann-Strebel invariants.

Free minimal actions of solvable Lie groups which are not affable

We construct an uncountable family of transversely Cantor laminations of compact spaces defined by free minimal actions of solvable groups, which are not affable and whose orbits are not quasi-isometric to Cayley graphs.

Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces

Let $ G $ be the group of orientation-preserving isometries of a rank-one symmetric space $ X $ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $ \Gamma \subset G $ on the boundary of $ X $, which is diffeomorphic to a sphere. When $ X $ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.

Local rigidity of certain solvable group actions on tori

<p style='text-indent:20px;'>In this paper, we study a local rigidity property of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb Z \ltimes_\lambda \mathbb R $\end{document}</tex-math></inline-formula> affine action on tori generated by an irreducible toral automorphism and a linear flow along an eigenspace. Such an action exhibits a weak version of local rigidity, i.e., any smooth perturbations close enough to an affine action is smoothly conjugate to the affine action up to constant time change.

Rigidity of Some Abelian-by-Cyclic Solvable Group Actions on $${\mathbb {T}}^N$$

In this paper, we study a natural class of groups that act as affine transformations of $${\mathbb {T}}^N$$ . We investigate whether these solvable, “abelian-by-cyclic," groups can act smoothly and nonaffinely on $${\mathbb {T}}^N$$ while remaining homotopic to the affine actions. In the affine actions, elliptic and hyperbolic dynamics coexist, forcing a priori complicated dynamics in nonaffine perturbations. We first show, using the KAM method, that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action, provided certain Diophantine conditions on the action are met. In dimension two, under natural dynamical hypotheses, we get a complete classification of such actions; namely, any such group action by $$C^r$$ diffeomorphisms can be conjugated to the affine action by $$C^{r-\varepsilon }$$ conjugacy. Next, we show that in any dimension, $$C^1$$ small perturbations can be conjugated to an affine action via $$C^{1+\varepsilon }$$ conjugacy. The method is a generalization of the Herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics.

Measure rigidity for solvable group actions in the space of lattices

We study invariant probability measures on the homogeneous space $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ for the action of subgroups of $\mathrm{SL}_n(\mathbb R)$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb R$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on $\mathrm{SL}_n(\mathbb R)/\mathrm{SL}_n(\mathbb Z)$ are homogeneous.

NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

Let $G$ be a finite group and $p$ a prime. We say that a $p$-regular element $g$ of $G$ is $p$-nonvanishing if no irreducible $p$-Brauer character of $G$ takes the value $0$ on $g$. The main result of this paper shows that if $G$ is solvable and $g\in G$ is a $p$-regular element which is $p$-nonvanishing, then $g$ lies in a normal subgroup of $G$ whose $p$-length and $p^{\prime }$-length are both at most 2 (with possible exceptions for $p\leq 7$), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for $p&gt;7$) some results in Dolfi and Pacifici [‘Zeros of Brauer characters and linear actions of finite groups’, J. Algebra 340 (2011), 104–113].

Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

This paper is the first part of a two part paper which introduces a method for determining the vanishing topology of nonisolated matrix singularities. A foundation for this is the introduction in this first part of a method for obtaining new classes of free divisors from representations V of connected solvable linear algebraic groups G. For equidimensional representations where dimG = dimV , with V having an open orbit, we give sufficient conditions that the complement E of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor). We do so by introducing the notion of a “block representation”which is especially suited for both solvable groups and extensions of reductive groups by them. This is a representation for which the matrix representing a basis of associated vector fields on V defined by the representation can be expressed using a basis for V as a block triangular matrix, with the blocks satisfying certain nonsingularity conditions. We use the Lie algebra structure of G to identify the blocks, the singular structure, and a defining equation for E. This construction naturally fits within the framework of towers of Lie groups and representations yielding a tower of free divisors which allows us to inductively represent the variety of singular matrices as fitting between two free divisors. We specifically apply this to spaces of matrices including m×m symmetric, skew-symmetric or general matrices, where we prove that both the classical Cholesky factorization of matrices and a further “modified Cholesky factorization”which we introduce are given by block representations of solvable group actions. For skew-symmetric matrices, we further introduce an extension of the method valid for a representation of a nonlinear infinite dimensional solvable Lie algebras. In part II, we shall use these geometric decompositions and results for computing the vanishing topology for nonlinear sections of free divisors to compute the vanishing topology for matrix singularities for all of the classes.

Zeros of Brauer characters and linear actions of finite groups

Let G be a finite group, and p a prime number greater than 3. It is known that, if every irreducible p-Brauer character of G does not vanish on any p ′ -element of G, then G is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, we show that the p ′ -length of G is at most 2 (the bound being the best possible). The structural results are obtained as an application of the main theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces.

Periodic points of solvable group actions on 1-arcwise connected continua

It is shown that every continuous action of a solvable group on a 1-arcwise connected continuum must have a fixed point or have a 2-periodic point.

Topological transitivity of solvable group actions on the line R

Let $\phi:G\rightarrow {\rm Homeo_+}(\mathbb{R})$ be an orientation preserving action of a discrete solvable group $G$ on $\mathbb R$. In this paper, the topological transitivity of $\phi$ is investigated. In particular, the relations between the dynamica

Large orbits in coprime actions of solvable groups

Let G be a solvable group of automorphisms of a finite group K. If |G| and |K| are coprime, then there exists an orbit of G on K of size at least √|G|. It is also proved that in a π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups.

Global rigidity of solvable group actions onS1

In this paper we find all solvable subgroups of Diff^omega(S^1) and classify their actions. We also investigate the C^r local rigidity of actions of the solvable Baumslag-Solitar groups on the circle. The investigation leads to two novel phenomena in the study of infinite group actions on compact manifolds. We exhibit a finitely generated group Gamma and a manifold M such that: * Gamma has exactly countably infinitely many effective real-analytic actions on M, up to conjugacy in Diff^omega(M); * every effective, real analytic action of Gamma on M is C^r locally rigid, for some r>=3, and for every such r, there are infinitely many nonconjugate, effective real-analytic actions of Gamma on M that are C^r locally rigid, but not C^(r-1) locally rigid.

Linearly Equivalent Actions of Solvable Groups

We determine the positive integers n for which there exist a solvable group G and two non-conjugate subgroups of index n in G that induce the same permutation character.

Properties of schemes inherited from orbits

In [9] the restriction functor was shown to be an equivalence between the categories of t for example, f induces isomorphisms of cohomology up to a given dimension if the same is true of the morphisms fr . However, the group action on S is used only to provide a covering of X by the closures of its orbits (provinces in the terminology of Sect. I), and the final version of the basic theorem is stated with the group action on X replaced by a family of closed sets having the relevant properties. Sections 2 and 3 are principally concerned with solvable group actions, in particular, with finding the fundamental group and cohomology groups of a scheme with a connected solvable group action. For example, in Section 2 it