Solvable Groups Research Articles

Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward

Long-range entanglement--the backbone of topologically ordered states--cannot be created in finite time using local unitary circuits, or equivalently, adiabatic state preparation. Recently it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases. Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots". First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class two (such as $D_4$ or $Q_8$). We show that after the measurement, the wavefunction always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome. Moreover, the clean quantum double ground state can be deterministically prepared via feedforward--gates which depend on the measurement outcomes. Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group. As a special case, this gives an explicit protocol to prepare twisted quantum double for all solvable groups. Third, we argue that certain topological orders, such as non-solvable quantum doubles or Fibonacci anyons, define non-trivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots. Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones. This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.

Finite quotients of ultraproducts of finite perfect groups

Groups that can be approximated by finite groups have been the subject of extensive research. This has led to the investigations of the subgroups of algebraic ultraproducts of finite groups, i.e., LEF groups. This paper addresses the dual problem: what are the abstract quotients of ultraproducts of finite groups? Certain cases were already well-studied. For example, if we have an ultraproduct of solvable group, then it is well-known that any finite abstract quotients must still be solvable.This paper studies the case of ultraproducts of finite perfect groups. We shall show that any finite group could be a quotient of an ultraproduct of finite perfect groups. We provide an explicit construction on how to achieve this for each finite group.

Periodic orbits of linear flows on connected Lie groups

Our main goal is to study the periodic orbits of linear flows on a real, connected Lie group. Since each linear flow φt has a derivation associated 𝒟, we show that the existence of periodic orbits of φt is based on the eigenvalues of the derivation 𝒟. From this, we study periodic orbits of a linear flow on noncompact, semisimple Lie groups, and we work with periodic orbits of a linear flow on a connected, simply connected, solvable Lie groups of dimension 2 or 3.

Regular and p-Regular Orbits of Solvable Linear Groups, II

Let [Formula: see text] be a faithful [Formula: see text]-module for a finite group [Formula: see text] and let[Formula: see text] be a prime dividing [Formula: see text]. An orbit [Formula: see text] for the action of [Formula: see text] on[Formula: see text] is regular if [Formula: see text], and is [Formula: see text]-regular if [Formula: see text]. In this note, we study two questions, one by the authors and one by Isaacs, related to the [Formula: see text]-regular orbits and regular orbits of the linear group actions.

Non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension

AbstractWe determine the finite non‐abelian simple groups which occur as the type of a Hopf–Galois structure on a solvable extension. In the language of skew braces, our result gives a complete list of finite non‐abelian simple groups which occur as the additive group of a skew brace with solvable multiplicative group.

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.

Using edt0l systems to solve some equations in the solvable Baumslag-Solitar groups

We investigate the solution sets to equations in the solvable Baumslag-Solitar groups BS(1,k), k≥2, and show that these sets are represented by edt0l languages in some cases. In particular, we prove that the multiplication table of such a group forms an edt0l language with respect to a specific natural normal form for group elements.

Left invariant Ricci flat metrics on Lie groups

Abstract In this paper, we apply the double extension process to study left invariant Ricci flat metrics on solvable and non-solvable Lie groups. An inductive method to produce new Ricci flat metrics from the old ones is established. As applications, we prove the following two results: (i) Every nilpotent Lie group with dim ⁡ C ⁢ ( G ) ≥ 1 2 ⁢ ( dim ⁡ G - 1 ) {\dim\mathrm{C}(G)\geq\frac{1}{2}(\dim G-1)} admits a left invariant Ricci flat metric. (ii) Given a Lie group G, there exists a nilpotent Lie group N with nilpotent index at most 2 such that G × N {G\times N} admits a left invariant Ricci flat metric. We also construct infinitely many new explicit examples of left invariant Ricci flat metrics on nilpotent Lie groups.

Solvable groups with four conjugacy classes outside a normal subgroup

Abstract In this paper, we investigate the structure of finite solvable groups G admitting a normal subgroup N such that G / N {G/N} is non-abelian and N contains all but four conjugacy classes of G.

Orders of products of elements and nilpotency of terms in the lower central series and the derived series

In this paper we prove that if G is a finite group, then the k -th term of the lower central series is nilpotent if and only if for every \gamma_{k} -values x,y \in G with coprime orders, either \pi(o(x)o(y))\subseteq \pi(o(xy)) or o(x)o(y) \leq o(xy) . We obtain an analogous version for the derived series of finite solvable groups, but replacing \gamma_{k} -values by \delta_{k} -values. We will also discuss the existence of normal Sylow subgroups in the derived subgroup in terms of the order of the product of certain elements.

Groups with fewer than 15 involutions

We show that if G is a finite group with fewer than 15 involutions, then . We then show that if G contains fewer than 15 involutions such that , then G is a solvable group of 2-rank 1. Furthermore, we show that if G contains involutions and , then G is a supersolvable group, but it is not necessarily true for . Finally, we show that a non-solvable group generated by involutions has at least 15 involutions.

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of mathfrak {z}-standard Sasaki solvable Lie algebras of dimension 2n+3, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify mathfrak {z}-standard Sasaki solvable Lie algebras of dimension le 7 and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

Measurable Foliations Associated to the Coadjoint Representation of a Class of Seven-Dimensional Solvable Lie Groups

We consider connected and simply connected seven-dimensional Lie groups whose Lie algebras have nilradical $\g_{5,2}$ of Dixmier. First, we give geometric descriptions of the maximal-dimensional orbits in the coadjoint representation of all considered Lie groups. Next, we prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. Finally, the topological classification of all these foliations is also provided.

On stably finiteness for $$C^*$$-algebras of exponential solvable Lie groups

We study the link between stably finiteness and stably projectionless-ness for $$C^*$$ -algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by 4; otherwise, they are not necessarily equivalent. To provide examples proving the last assertion, we study exponential solvable Lie groups that have nonempty finite open sets in their unitary dual.

On p′-degree ℓ-Brauer characters of ℓ-solvable groups

Let [Formula: see text] and [Formula: see text] be distinct primes, let [Formula: see text] be a finite [Formula: see text]-solvable group and let [Formula: see text]. Then [Formula: see text] has a unique irreducible [Formula: see text]-Brauer character of [Formula: see text]-degree lying over [Formula: see text] if and only if NG (P)/ P is an [Formula: see text]-group. This extends a result from [J., F.,Tent, Correspondences of Brauer characters and Sylow subgroup normalizers, J. Algebra 573 (2021) 436–450] to every odd prime [Formula: see text].

Compatibility of Balanced and SKT Metrics on Two-Step Solvable Lie Groups

AbstractIt has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible Kähler metric. Using the shear construction and classification results for two-step solvable SKT Lie algebras from our previous work, we prove this conjecture for compact two-step solvmanifolds endowed with an invariant complex structure which is either (a) of pure type or (b) of dimension six. In contrast, we provide two counterexamples for a natural generalisation of this conjecture in the homogeneous invariant setting. As part of the work, we obtain further classification results for invariant SKT, balanced and Kähler structures on two-step solvable Lie groups. In particular, we give the full classification of left-invariant SKT structures on two-step solvable Lie groups in dimension six.

Finite solvable groups with two imprimitive monolithic characters

Let [Formula: see text] be a finite group and let [Formula: see text] be an irreducible character of [Formula: see text]. If [Formula: see text] has a unique minimal normal subgroup, then [Formula: see text] is called monolithic. The character [Formula: see text] is said to be imprimitive if [Formula: see text] is induced from a character of a proper subgroup of [Formula: see text]. In this paper, we classify the finite solvable groups having exactly two imprimitive monolithic characters whose kernels intersect trivially.

Classifying solvable primitive permutation groups of low rank

Classifying solvable primitive permutation groups of low rank

The BNS invariants of the generalized solvable Baumslag-Solitar groups and of their finite index subgroups

We compute the Bieri-Neumann-Strebel invariants for the generalized solvable Baumslag-Solitar groups and their finite index subgroups. Using , we show that certain finite index subgroups of cannot be isomorphic to for any k. In addition, we use the BNS-invariants to give a new proof of property for the groups and their finite index subgroups.

Coherence of augmented Iwasawa algebras

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.