Envelopes of certain solvable groups

Higher rank confining subsets and hyperbolic actions of solvable groups

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Let F be a discrete uniform subgroup of a connected simply connected solvable Lie group S. It is shown how S is essentially determined by F, using the point of view of group extensions. Let r be a discrete uniform subgroup of a connected simply connected solvable Lie group S. The main purpose of this work is to study how S is essentially determined by T from the point of view of group extensions. In [5], Mal'cev proved that T determines 5 uniquely if S is nilpotent. However, for a general solvable group, the situation seems less favorable. Nevertheless, L. Auslander (Theorem 2 in [1]) has obtained some results when T is the fundamental group of a nilmanifold, and in the subsequent works of Auslander and Tolimieri ([2], [8], [9]), stronger results were proved by using the notation of semisimple splittings, from which one can obtain the generalization of the result in [1] to arbitrary discrete uniform subgroups T. It is the purpose of this paper to revisit and prove directly this generalization by making use of the theory of group extensions as Auslander originally did in [1]. The author is indebted to the referee for pointing out the above results of Auslander and Tolimieri. 1. Let G and H be topological groups. An extension of G by H is a pair (E, rr) consisting of a topological group E which contains G as a closed normal subgroup and a continuous open homomorphism it of E onto H whose kernel is G. Two extensions (Et, 7r,-) of G by H are said to be equivalent if there exists an isomorphism a : Ex-+E2 of topological groups which leaves elements of G fixed and is such that tt2o=ttx. If (E, it) is an extension of G by H, then this determines a homomorphism Tr°:H—>0(G), where 0(G) denotes the group A(G) of all automorphisms of G modulo the inner automorphism group 1(G) of G. If (Ex, irx) and (E2, tt2) are equivalent, then 77^=712. For any homomorphism -0(G), let Ext(G, H, cp) denote the set of all equivalence classes of the extensions (E, it) of G by H Received by the editors July 12, 1971 and, in revised form, November 17, 1971. AMS 1970 subject classifications. Primary 22E15, 22E20, 22E40.

The well-known theory of square-integrable representations is generalized to the case of primary representations (not necessarily type I) quasi-contained in either the regular representation or the representation induced from a character of the center of a (not necessarily unimodular) locally compact group, and relations with the topology of the primitive ideal space of the group C*-algebra are obtained.The cases of discrete and almost connected groups are examined in more detail, and it is shown that for such groups, square-integrable factor representations must be traceable.For connected Lie groups, these representations can (in principle) be determined up to quasi-equivalence using a complicated construction of L. Pukanszky-for type I simply connected solvable Lie groups, the characterization reduces to that conjectured by C. C. Moore and J. Wolf.In the case of unimodular exponential groups, essentially everything is as in the nilpotent case (including a result on multiplicities in the decomposition of L2(G/T), T a discrete uniform subgroup of G).Finally, it is shown that the same criterion as for type I solvable Lie groups characterizes the squareintegrable representations of certain solvable p-adic groups studied by R. Howe.1. Introduction.The theory of square-integrable irreducible representations of unimodular groups has been understood for some time, and is easily accessible in [8, 14] and [36].Modifications have been introduced to take into account groups with noncompact center (e.g., [24]) and nonunimodular groups (most notably [10]; cf. also [25], [26]), but until recently, we were unaware of any treatment of non-type I square-integrable representations aside from some remarks on cyclic representations in [36], [25], and [26].A few months ago, the author received a copy of [7], which, in Chapter III, does treat a much more general situation, but here the emphasis is on von Neumann algebras and not much is said about groups.The lack of study of general square-integrable representations would not be much of a deficiency were it not for the fact that many discrete groups and Lie groups have

Let GL(n, R) denote the real general linear group with the usual topology. A subgroup of GL(n, R) will be called an w-matrix group. We will say that a subgroup G of GL(n, R) is a discrete group if it is discrete in the induced topology. If for any group G, [G, G] denotes the commutator subgroup of G, we will say that a group 5 is solvable provided the sequence of groups S = Slt [Si, Si] = S2, • • • , [Sic-i, Sk-i]=Sk, terminates in the identity element for some k. If Sk = e and Sk-i^e, where e is the identity in S, then we will say that S is k step solvable and k is called also the index of solvability. Then in [l] and [4] we have independent proofs of the following theorem:

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\Z_{p}^{n}$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

We consider distributions of dislocations in continuum models of crystals which are such that the corresponding dislocation density tensor relates to a particular class of solvable Lie group, and discrete structures which are embedded in these crystals. We provide a canonical form of these structures and, by finding the set of all generators of a corresponding discrete subgroup, we determine the ‘material’ symmetries that constrain appropriate strain energy functions.

The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are(i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and(ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.

Hardy’s uncertainty principle for the Gabor transform is proved for locally compact abelian groups having noncompact identity component and groups of the form$\mathbb{R}^{n}\times K$, where$K$is a compact group having irreducible representations of bounded dimension. We also show that Hardy’s theorem fails for a connected nilpotent Lie group$G$which admits a square integrable irreducible representation. Further, a similar conclusion is made for groups of the form$G\times D$, where$D$is a discrete group.

In Pontrjagin's theory of duality for compact abelian groups, the following theorem is known : Let G be a compact abelian group, G* the dual group. Then the topological dimension of G, in the sense of Lebesgue, is equal to the rank of discrete abelian group G*. It was Prof. T. Tannaka who has called my attention to the lack of corresponding theorem in non-commutative case. I intend to give, in this note, a theorem of this kind in the following form: THEOREM A. Let G be an arbitrary compact group, G the aggregate of continuous finite dimensional representstions of G, C[ G ] the algebra over the complex numbers C generated by the coefficients of representations in G, i.e., the representative ring of G in the sense of C. Chevalley^. Then the topological dimension of G, in the sense of Lebesgue, is equal to the transcendental degree of C[G] over C.

In this paper, we study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of [P. Hall and C. R. Kulatilaka, A property of locally finite groups, J. Lond. Math. Soc. 39 1964, 235–239] and a characterization of a certain class of Lie groups, due to [S. K. Grosser and W. N. Herfort, Abelian subgroups of topological groups, Trans. Amer. Math. Soc. 283 1984, 1, 211–223], we prove that a c-minimal locally solvable Lie group is compact. It is shown that a topological group G is c-(totally) minimal if and only if G has a compact normal subgroup N such that G / N G/N is c-(totally) minimal. Applying this result, we prove that a locally compact group G is c-totally minimal if and only if its connected component c ⁢ ( G ) c(G) is compact and G / c ⁢ ( G ) G/c(G) is c-totally minimal. Moreover, a c-totally minimal group that is either complete solvable or strongly compactly covered must be compact. Negatively answering [D. Dikranjan and M. Megrelishvili, Minimality conditions in topological groups, Recent Progress in General Topology. III, Atlantis Press, Paris 2014, 229–327, Question 3.10 (b)], we find, in contrast, a totally minimal solvable (even metabelian) Lie group that is not compact.

Let G G be a locally compact group P ( G ) = { 0 ≦ ϕ ∈ L 1 ( G ) ; ∫ ϕ ( x ) d x = 1 } P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} } and ( l a f ) ( x ) = a f ( x ) = f ( a x ) ({l_a}f)(x) = {}_af(x) = f(ax) for all a , x ∈ G a,x \in G and f ∈ L ∞ ( G ) .0 ≦ Ψ ∈ L ∞ ( G ) ∗ , Ψ ( 1 ) = 1 f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1 is said to be a [topological] left invariant mean ([TLIM] LIM) if Ψ ( a f ) = Ψ ( f ) [ Ψ ( ϕ ∗ f ) = Ψ ( f ) \Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f) ] for all a ∈ G , ϕ ∈ P ( G ) , f ∈ L ∞ ( G ) a \in G,\phi \in P(G),f \in {L^\infty }(G) . The main result of this paper is the Theorem. Let G G be a locally compact group, amenable as a discrete group. If G G contains an open σ \sigma -compact normal subgroup, then LIM = TLIM if and only if G G is discrete. In particular if G G is an infinite compact amenable as discrete group then there exists some Ψ ∈ L I M \Psi \in LIM which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper . 2 ^{2}

In this second paper we continue our development of elementary methods for computing continuous complex-valued solutions of the 2-cocycle functional equation on locally compact groups. The first paper dealt primarily with solvable groups. Here we consider some classical motion groups of mathematical physics, including the Euclidean, Galilean, Lorentz and Poincare groups.