Solvable Actions Research Articles

Frames arising from irreducible solvable actions I

In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time–frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra g=p+m. Next, let π be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup P=exp⁡p of G. Additionally, we assume that G=P⋊M, M=exp⁡m is a closed subgroup of G, dμM is a fixed Haar measure on the solvable Lie group M and there exists a linear functional λ∈p⁎ such that the representation π=πλ=indPG(χλ) is realized as acting in L2(M,dμM). Making no assumption on the integrability of πλ, we describe explicitly a discrete subset Γ of G and a vector f∈L2(M,dμM) such that πλ(Γ)f is a tight frame for L2(M,dμM). We also construct compactly supported smooth functions s and discrete subsets Γ⊂G such that πλ(Γ)s is a frame for L2(M,dμM).

Cohomology of linking systems with twisted coefficients by a $p$-solvable action

In this paper we study the cohomology of the geometric realization of linking systems with twisted coefficients. More precisely, given a prime $p$ and a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, we compare the cohomology of $\mathcal{L}$ with twisted coefficients with the submodule of $\mathcal{F}^c$-stable elements in the cohomology of $S$. We start with the particular case of constrained fusion systems. Then, we study the case of $p$-solvable actions on the coefficients.

Rigidity of certain solvable actions on the sphere

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere by conformal transformations. The action is not locally rigid in higher dimension, but exhibits a weak form of local rigidity. More precisely, any perturbation preserves a smooth conformal structure.

Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, 1 → F → G → H → 1 1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1 , we prove that G G has solvable conjugacy problem if and only if the corresponding action subgroup A ⩽ A u t ( F ) A\leqslant Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form Z 2 ⋊ F m \mathbb {Z}^2\rtimes F_m , F 2 ⋊ F m F_2\rtimes F_m , F n ⋊ Z F_n \rtimes \mathbb {Z} , and Z n ⋊ A F m \mathbb {Z}^n \rtimes _A F_m with virtually solvable action group A ⩽ G L n ( Z ) A\leqslant GL_n(\mathbb {Z}) . Also, we give an easy way of constructing groups of the form Z 4 ⋊ F n \mathbb {Z}^4\rtimes F_n and F 3 ⋊ F n F_3\rtimes F_n with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and we give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in A u t ( F 2 ) Aut(F_2) is given.